Optical Excitation of Surface
Plasmon Polaritons on Novel
Bigratings
Thomas James Constant
School of Physics
University of Exeter
A thesis submitted for the degree of
Doctor of Philosophy
March 2013
Optical Excitation of Surface Plasmon
Polaritons on Novel Bigratings
Submitted by Thomas James Constant to the University of Exeter as a thesis for
the degree of Doctor of Philosophy in Physics
2013
This thesis is available for Library use on the understanding that it is copyright
material and that no quotation from the thesis may be published without proper
acknowledgement.
I certify that all material in this thesis which is not my own work has been identified
and that no material has previously been submitted and approved for the award of a
degree by this or any other University.
Thomas James Constant
2013
For my mum, Lel, and my sister, Emma.
Abstract
This thesis details original experimental investigations in to the interaction of light with
the mobile electrons at the surface of metallic diffraction gratings. The gratings used in
this work to support the resultant trapped surface waves (surface plasmon polaritons),
may be divided into two classes: ‘crossed’ bigratings and ‘zigzag’ gratings.
Crossed bigratings are composed of two diffraction gratings formed of periodic
grooves in a metal surface, which are crossed at an angle relative to one another. While
crossed bigratings have been studied previously, this work focuses on symmetries which
have received comparatively little attention in the literature. The gratings explored in
this work possesses two different underlying Bravais lattices: rectangular and oblique.
Control over the surface plasmon polariton (SPP) dispersion on a rectangular
bigrating is demonstrated by the deepening of one of the two constituent gratings.
The resulting change in the diffraction efficiency of the surface waves leads to large
SPP band-gaps in one direction across the grating, leaving the SPP propagation in the
orthogonal direction largely unperturbed. This provides a mechanism to design surfaces
that support highly anisotropic propagation of SPPs.
SPPs on the oblique grating are found to mediate polarisation conversion of the
incident light field. Additionally, the SPP band-gaps that form on such a surface are
shown to not necessarily occur at the Brillouin Zone boundaries of this lattice, as the
BZ boundary for an oblique lattice is not a continuous contour of high-symmetry points.
The second class of diffraction grating investigated in this thesis is the new zigzag
grating geometry. This grating is formed of sub-wavelength (non-diffracting) grooves
that are ‘zigzagged’ along their length to provide a diffractive periodicity for visible
frequency radiation. The excitation and propagation of SPPs on such gratings is
investigated and found to be highly polarisation selective.
The first type of zigzag grating investigated possesses a single mirror plane. SPP
excitation to found to be dependant on which diffracted order of SPP is under polarised
illumination. The formation of SPP band-gaps is also investigated, finding that the
band-gap at the first Brillouin Zone boundary is forbidden by the grating’s symmetry.
The final grating considered is a zigzag grating which possesses no mirror symmetry.
Using this grating, it is demonstrated that any polarisation of incident light may
resonantly drive the same SPP modes. SPP propagation on this grating is found to be
forbidden in all directions for a range of frequencies, forming a full SPP band-gap.
Acknowledgements
First and foremost, I would like to thank my supervisors, Professor Roy Sambles and
Professor Alastair Hibbins. Roy has been a constant inspiration to me throughout my
research, and has kept me motivated and excited in all that I have done. The level of
care and diligence he affords his students is second to none. Thank you Roy. In Alastair,
I have gained not just a respected colleague, but also a close friend. In the early days of
my supervisory meetings, Al was always there to step in and say “Hang on, Roy, this
isn’t trivial.”, for which I will be forever grateful. I’ve also enjoyed many fruitful hours
with Al at conferences, and many less fruitful hours down the pub. Thank you Al.
Many, many members of the Electromagnetic Materials Group at Exeter have helped
me throughout my Ph. D. None more so than Matt Lockyear, without whose advice and
support I most certainly wouldn’t have made it this far. Matt and I was fortunate enough
to be part of RAD crew during it’s golden age, a rag-tag bunch of physicist-surfers who
had some rather fantastic adventures.
Ian Hooper is Exeter’s resident grating expert, and I am indebted to him for countless
discussions, arguments and comments on the physics of the diffraction grating. Currently,
our running pool score stands at 97-65 to Hoops, and by the time anyone reads this I
am certain he will have reached 100 and I will have had to buy us a holiday.
In the first few years of my work, I was lucky enough to have Professor Alex
Savchenko as my academic mentor. Alex helped me a great deal when I was struggling,
and was one of the most down to earth, empathic people I’ve ever met. Even after
his untimely death in 2011, Alex’s legacy continues to have an important effect on my
research. Alex’s winning of the funding for Exeter’s graphene center brought reliable
nano-fabrication technology and immeasurably skilled physicists to Exeter, and in doing
so allowed me to create the samples presented in this thesis.
I must thank Dave Anderson for his supreme skill at electron beam lithography and
his remarkably patient and helpful attitude in dealing with me. A lot of the samples
used in this work simply could not have been made without Dave’s help.
Plenty of the other academics have been great fun to work with at Exeter. Pete
Vukusic has been a great help to me, providing me in the final months with some
funding to develop the scatterometry kit further. Bill Barnes has always provided
thought provoking questions, and the back-to-basics sessions he has run helped me
lots with, well, the not-so-basics of our field. Euan Hendry has also provided useful
discussions, and has also wiped me out many times in poker. A four hour chat we had
about Fourier series in optics, whilst sharing some beers in the baggage compartment of
a train from London to Exeter, still helps me heavily in the physical reasoning in this
thesis.
The technical workshop have been fantastic in my time at Exeter, making some
remarkable devices and some less remarkable Bar stands. Originally my go-to man was
Pete Cann, and after he left us to go enjoy Australia and then later family, the burden
then fell to Nick Cole. They are both fantastic technicians and have made many of
the results in the thesis possible by providing, or sometimes just fixing, vital kit. The
other workshop boys have also been great. John, Kev and Adam have all helped me
greatly, be it helping with orders (John), making things Nick can’t (Kev), or just yelling
instructions at me as our star defender on the post-grad football team (Adam!). In the
techinical areas, I can’t forget Chris Forrest, who is often a lifesaver with all things IT.
Team Basement, as the dwellers of our dankest subterranean lab named ourselves,
were constant fun. Ed Stone, for sharing and increasing my love for R, introducing me to
Linux and providing some of the most memorable moments of these past years, I thank
you! Earthquake testing Ed, though ill-advised, is possibly one of the funniest things
you’ll ever see. Caroline Pouya has managed to put up with me now for close to nine
years. How she’s managed that is completely beyond me, and despite such prolonged
exposure to me, she has become one of my closest friends. Later, we were joined by Alfie,
Nixon and in the final days Luke and Tim, who all took the challenge of living without
daylight and the constant fear of asphyxiation, electrocution or asbestos poisoning with
the good humour we’ve come to expect from Team Basement. I’d also like to thank
Ed and Alfie particularity for listening to my ideas about using scatterometry to map
iso-frequency contours, and helping me develop the experiential technique which is used
heavily in this thesis. Chris Holmes was also there, but we shall get to him later.
The microwave kids in G31 have also been wonderful friends. Celia Butler has always
been there for a chat, and I certainly now know more about Brownies and Gambia
than I ever intended to. Mel Taylor and I did some summer project work together just
prior to the Ph. D., and we had some great fun. Later Al, Simon, Panda, Liz, Laura,
Ben and Ruth joined us, and they’re all been fantastic friends. Another microwave
bunny, Helen Rance is (quite by accident) my oldest friend at Exeter. We’ve known
each other for so long, and through so many different experiences, that to attempt a
decent acknowledgement is an exercise in futility.
For the majority of my Ph. D., I lived in Horseguards parade with Steve Hubbard,
Matt Biginton, Ciar´an Stewart, Chris Holmes and Pete Hale. Stormguards, as we
affectionately named it, was not just a house. It was an exclusive gentleman’s club, a
brewery, a philosophical forum, a base of operations, a darts club, a support group, a
fraternity and a state of mind. The golden age of the Stormers saw us invent the amateur
beer festival, the glove game, tea-darts, Nakatomi Plaza, develop an appreciation of fine
whisky and much much more. I think it’s fair to say that the two key players (other
than myself) in the development of Storm-Culture was Pete and Chris; so Stormers, I
thank you especially.
Plenty of other physicists have been important to me throughout this work. To list
a few, I must mention Nat, Babs, Laureline, Tom D, and not forgetting the Russian
contingent in Tim, Ivan, and the rest of the quantum group.
I have been fortunate to have many great friends outside of the department over the
past 8 years. Holly Keatings and I met on the second day of uni, and have together
sampled almost every restaurant in Exeter. My other house-mates of that era were
Ashley, Rob and Jon. It was my great privilege to see Lash marry a fellow Stormer,
Steve, after introducing them at our house while making name badges.
For many years I competed for the university as part of the fencing club. On the
Men’s 1
st
team I met Graham Heydon and and James Parker, who are now two of my
best friends. The adventures and trouble we got ourselves in make any attempt to play
‘I have never’ with these guys total suicide. I should also mention Pippa, otherwise
she’ll get in a huff.
My Aunt, Terri and my Uncle, Joe have let me live with them for the past 5 months,
and have let me escape the city to be able to write this thesis. They have made me feel
exceptionally welcome and tremendously loved during my stay. I’m lucky enough to say
that this affection and kindness isn’t unique to this period of writing, but is the norm
for my entire life. Thank you, Terri and Joe.
Finally I must thank Emma and Mum. My little sister Em has always been
encouraging and interested in what I’ve been doing, and as she’s grown up has become
one of my best friends. My mum, Lel, apparently used to flash brightly coloured cards
at me while I was a baby in my cot, hoping to somehow induce intelligence. I’m not
sure that worked, but her wish, motivation and encouragement for me to succeed at
anything I put my mind to is pretty much a running theme of my entire life. It is
certainly why I get to be writing this today. I love them both very much.
Contents
Contents vi
List of Figures ix
1 Introduction 1
1.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Scope and Outline of This Work . . . . . . . . . . . . . . . . . . . . . . 4
2 Background Theory 7
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Surface Polaritons on Planar Surfaces . . . . . . . . . . . . . . . . . . . 7
2.3 Dispersion of SPPs on Planar Surfaces . . . . . . . . . . . . . . . . . . . 11
2.3.1 Penetration Depth . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.2 Propagation Length . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Surface Plasmon Polaritons on Diffraction Gratings . . . . . . . . . . . . 16
2.4.1 Diffraction Gratings . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.2 Coupling Surface Plasmons to Light by Coherent Scattering . . . 18
2.4.3 Surface Plasmon Polaritons on Bigratings . . . . . . . . . . . . . 23
2.4.4 Coupling Strength of Light to SPPs on Gratings . . . . . . . . . 25
2.4.5 Plasmonic Band Gaps . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4.5.1 Coupling Light to Band Edges . . . . . . . . . . . . . . 29
2.4.6 Polarisation Conversion . . . . . . . . . . . . . . . . . . . . . . . 31
3 Theoretical Methods 33
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 The Differential Method of Chandezon et al. . . . . . . . . . . . . . . . . 34
3.3 The Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3.1 Solving Maxwell’s Equations in the Mesh . . . . . . . . . . . . . 36
3.3.2 Adaptive Meshing . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3.3 Solution Evaluation and Convergence . . . . . . . . . . . . . . . 37
vi
CONTENTS
3.3.4 Model Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 37
3.3.5 Floquet Type Excitation . . . . . . . . . . . . . . . . . . . . . . . 39
4 Experimental Methodology 41
4.1 Sample Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.1.1 Electron Beam Lithography . . . . . . . . . . . . . . . . . . . . . 41
4.1.2 Thermal Evaporation . . . . . . . . . . . . . . . . . . . . . . . . 45
4.1.3 Template Stripping . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.1.4 Polymer Replication . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2 Angle Scans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2.1 Monochromator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2.2 Fixed Wavelength Scans . . . . . . . . . . . . . . . . . . . . . . . 48
4.2.3 Embedded Samples . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.3 Iso-Frequency Contour Measurement Using Scatterometry . . . . . . . 49
4.3.1 Scatterometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3.2 Principle of Operation . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3.3 Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3.4 The Role of Polarisation . . . . . . . . . . . . . . . . . . . . . . . 54
4.3.5 Corrections and Momentum-Space Deformation . . . . . . . . . . 55
4.3.6 Example Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5 Optical Response of Metallic Rectangular Bigratings 58
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.2 The Rectangular Bigrating . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.3 Dispersion of SPPs on Rectangular Bigratings . . . . . . . . . . . . . . . 60
5.3.1 Scattering Components on Lamellar Gratings . . . . . . . . . . . 60
5.3.2 Experimental Mapping of SPP Dispersion . . . . . . . . . . . . . 62
5.3.3 Band Gap Observation Using Scatterometry . . . . . . . . . . . . 66
5.4 Polarisation Conversion on Rectangular Symmetry . . . . . . . . . . . . 68
5.5 Controlling SPP Anisotropy Using Rectangular Bigratings . . . . . . . . 71
5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6 Optical Response of Metallic Oblique Bigratings 77
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.2 The Oblique Grating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.3 Coupling of Light and SPP Mode Interaction on Oblique Gratings . . . 80
6.4 Polarisation Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.5 SPPs at the BZ Boundary of Oblique Bigratings . . . . . . . . . . . . . 85
6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
vii
CONTENTS
7 Optical Response of Metallic Zigzag Bigratings 92
7.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7.2 The Zigzag Grating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7.3 The Coupling of Plane Polarised Light to SPPs on Zigzag Gratings . . . 94
7.3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
7.3.2 Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.3.3 Transverse Electric Coupling . . . . . . . . . . . . . . . . . . . . 100
7.3.4 Transverse Magnetic Coupling . . . . . . . . . . . . . . . . . . . 104
7.4 Band Structure of SPPs on a Zigzag Grating . . . . . . . . . . . . . . . 107
7.4.1 Band-Gaps at k
x
= 0 . . . . . . . . . . . . . . . . . . . . . . . . . 108
7.4.2 Band-Gaps at the First Brillouin Zone Boundary . . . . . . . . . 112
7.5 Anisotropic Propagation of SPP Modes for Self Collimation . . . . . . . 115
7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
8 Optical Response of Asymmetric Zigzag Bigratings 122
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
8.2 The Asymmetric Zigzag Grating . . . . . . . . . . . . . . . . . . . . . . 123
8.3
The Coupling of Polarised Light to SPPs on an Asymmetric Zigzag Grating
126
8.3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
8.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
8.4 Band Structure of SPPs on an Asymmetric Zigzag Grating . . . . . . . 134
8.4.1 Band Gaps at k
x
= 0 . . . . . . . . . . . . . . . . . . . . . . . . . 135
8.4.2 Band Gaps at the 1
st
BZ Boundary . . . . . . . . . . . . . . . . 136
8.4.2.1 Coupling of Light to the Band Edges . . . . . . . . . . 138
8.4.2.2 A Full Surface Plasmon Band Gap . . . . . . . . . . . . 141
8.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
9 Conclusions 145
9.1 Summary of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
9.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
Publications 150
References 152
viii
List of Figures
1.1
The four types of grating geometry that are investigated in this thesis:
The rectangular bigrating, the oblique bigrating, the zigzag grating, and
the asymmetric zigzag grating. . . . . . . . . . . . . . . . . . . . . . . . 4
2.1
A schematic representation of propagating electromagnetic fields at an
interface between two materials. . . . . . . . . . . . . . . . . . . . . . . 9
2.2
The Drude model for silver calculated with
ω
p
= 1
.
32
×
10
16
Hz, γ
=
1.4 × 10
14
Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3
The dispersion of a surface plasmon polariton on a planar film approxi-
mated with the Drude mode. . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4
Diagram of the electric field vectors of a SPP at the interface between a
metal and a dielectric and the exponential decay of the
E
z
component
away from the surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 The coordinate system used for a simple grating in the conical mount. . 17
2.6
The allowed real momentum states for light incident on three diffraction
gratings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.7
Two dispersion diagrams illustrating in-plane diffractive coupling to
the zero-order SPP in the extended zone scheme, and the higher-order
diffracted SPPs couling to zero-order light in the reduced zone scheme . 21
2.8
3D plots demonstrating intersections of the plane of incidence with a
scattered SPP cone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.9
The reciprocal space map for an oblique bigrating with grating vectors
k
gx
and k
gv
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.10
An example spectrum for a simple sinusoidal silver grating, calculated
using the Chandezon method. . . . . . . . . . . . . . . . . . . . . . . . . 25
2.11 Cartoon of two possible SPP standing waves on a sinusoidal grating. . . 27
2.12
The dispersions and band gaps of a SPP on a grating with one Fourier
Harmonic and on a grating with the first two Fourier Harmonics. . . . . 28
ix
LIST OF FIGURES
2.13
Sketches of different grating profiles determined by the relative phases
of the
k
g
and 2
k
g
components, and the resulting coupling of the light to
the band-edges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.1
A graph showing the convergence of a typical FEM model as a function
of adaptive passes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2 Schematic of the different boundaries used in the FEM model. . . . . . 39
4.1
Illustration of the fabrication method used for the production of diffraction
gratings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2
System for the measurement of the reflectivity of a sample as a function
of both wavelength, λ
0
and polar angle θ. . . . . . . . . . . . . . . . . . 47
4.3
System for the measurement of angular dependant reflectivity at a set
wavelength of λ
0
= 632.8 nm . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.4 Exploded view of the sample mount for use with a glass hemisphere. . . 49
4.5
Three iso-frequency contours for the intersection of two SPP cones at
angular frequencies ω
1
, ω
2
and ω
3
. . . . . . . . . . . . . . . . . . . . . . 50
4.6
Experimental arrangement for the modified imaging scatterometry system.
53
4.7
Four raw images from the scatterometer for a wavelength of
λ
0
= 650 nm
with different polarisations. . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.8 The image correction for R(θ, φ) R(k
0
sin θ, φ). . . . . . . . . . . . . . 56
4.9
The processing of a raw dataset from the scatterometer at a wavelength
of λ
0
= 650 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.1 The coordinate system for a rectangular grating. . . . . . . . . . . . . . 60
5.2
Three examples of unit cells for lamellar grating profiles, and the square
of the Fourier coefficients, a
n
, for these groove profiles. . . . . . . . . . . 62
5.3
Scanning electron micrographs of the template-stripped rectangular bi-
grating in silver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.4
Reflectivity for different polarisations and azimuthal angles for a rectan-
gular grating mapped as a function of (ω, k). . . . . . . . . . . . . . . . 64
5.5
A scattergram of a rectangular grating mapped to
k
-space for an illumi-
nation wavelength of λ
0
= 550 nm. . . . . . . . . . . . . . . . . . . . . . 66
5.6
Cartoon of the (
1
,
0) and (1
,
0) SPP iso-frequency contours at the BZ
boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.7
Plots showing the electric field for the high energy and low energy SPP
standing waves occurring at the BZ boundary. . . . . . . . . . . . . . . . 68
5.8 Spectra of polarisation conversion on the rectangular grating at φ = 45
. 70
x
LIST OF FIGURES
5.9
Diagram of the effect on the surface profile by increasing the depth
d
2
of
the rectangular bigrating. . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.10
A sketch showing the expected iso-frequency contour deformation as
d
2
is changed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.11
Experimental iso-frequency contours for two rectangular bigratings at a
wavelength of λ
0
= 700 nm. . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.12
Experimentally obtained dispersion diagrams mapped from the reflectance
of rectangular bi-gratings with nominal depths of 40 nm and 80 nm . . . 75
6.1 Coordinate system for an oblique bigrating. . . . . . . . . . . . . . . . . 78
6.2 Example reciprocal lattices for an oblique grating. . . . . . . . . . . . . 79
6.3 SEM of an oblique grating master fabricated in a silicon wafer. . . . . . 80
6.4
The modelled and experimental iso-frequency surface for an oblique
bigrating illuminated with TM polarisation at λ
0
= 700 nm. . . . . . . . 81
6.5 The dispersion of modes on an oblique grating for φ = α = 75
. . . . . . 83
6.6 Polarisation conservation and conversion mediated by out-of-plane scat-
tered SPPs on an oblique grating. . . . . . . . . . . . . . . . . . . . . . . 85
6.7
An experimentally obtained iso-frequency contour map of SPPs on an
oblique bigrating with illuminated with incident light of wavelength
λ
0
= 650 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.8 Regions of interest from figure 6.7. . . . . . . . . . . . . . . . . . . . . . 87
6.9 Two possible primitive unit cells for an oblique lattice. . . . . . . . . . . 89
6.10
Diagram of the cancellation of a vector field on a rectangular and oblique
BZ boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.11 Dispersion plots mapped using the reflectivity of an oblique grating. . . 91
7.1 The coordinate system of a zigzag grating. . . . . . . . . . . . . . . . . . 94
7.2
The surface normal components of electric field vector for TM and TE
polarised light projected on a contour of the zigzag surface profile. . . . 95
7.3 The magnitude of the surface normal electric field in the x-direction for
TM and TE polarisations. . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7.4 Schematic cartoon of light coupling to SPPs on a zigzag grating. . . . . 98
7.5 Scanning electron micrographs of various zigzag grating samples. . . . . 99
7.6 SPP dispersion on a zigzag grating measured using TE polarised light. . 100
7.7
Plot of TE and TM polarised light reflectivity from the grating as a
function of polar angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.8 The E field plot of a SPP excited by TE radiation. . . . . . . . . . . . . 102
xi
LIST OF FIGURES
7.9
SPP dispersion on a zigzag grating in glass measured using TE polarised
light. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.10
An example experimental reflectivity plot for TE and TM polarised light
in the visible range at a fixed polar angle of θ = 53
. . . . . . . . . . . . 105
7.11
An example spectral plot and fit at
θ
= 53
for the TM reflectivity
normalised to the TE reflectivity, and comparison to theory. . . . . . . . 106
7.12
SPP dispersion on a zigzag grating in glass measured using TM polarised
light. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7.13
An iso-frequency scattergram mapping
k
-space contours for a 600 nm
zigzag grating for an energy of 2.14 eV . . . . . . . . . . . . . . . . . . . 108
7.14
Cartoon of the two standing wave solutions for SPPs at
k
x
= 0, projected
onto the zigzag surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7.15 The magnitude of electric field for the SPP standing waves at k
x
= 0. . 110
7.16
Manipulation of the band-gap observed at
k
x
= 0 through the increasing
of zigzag amplitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.17
Modelled SPP dispersion around the intersection point of the
1
k
gx
and
+2k
gx
scattered SPPs meeting at the first BZ boundary. . . . . . . . . . 112
7.18 Cartoon of the two standing wave solutions for SPPs at the 1
st
BZ. . . . 113
7.19
The magnitude of electric field,
|E|
for the degenerate SPP standing
waves at the first BZ. (xy plane) . . . . . . . . . . . . . . . . . . . . . . 114
7.20
The magnitude of electric field for the degenerate SPP standing waves at
the first BZ. (xy and xz plane) . . . . . . . . . . . . . . . . . . . . . . . 115
7.21
Experimentally obtained dispersion plots for a zigzag grating at
φ
= 0
and φ = 90
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.22
Measured iso-frequency contours of a zigzag grating for a range of wave-
lengths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7.23
The SPP iso-frequency contours mapped using imaging scatterometery
at λ
0
= 450 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
8.1 Coordinate system for the asymmetric zigzag grating. . . . . . . . . . . 124
8.2 Two possible unit cells for a zigzag grating: symmetric and asymmetric. 125
8.3
Scanning electron micrographs of an asymmetric zigzag silicon master
and the template stripped sample in silver. . . . . . . . . . . . . . . . . 125
8.4
A schematic of the electric field lines in the grooves of an asymmetric
zigzag grating for two polarisation cases. . . . . . . . . . . . . . . . . . . 127
8.5
The piecewise function representing
E
x
for TE (
E
T E
x
) polarisation and
TM (E
T M
x
) polarisation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
xii
LIST OF FIGURES
8.6
Magnitude of the electric field in region 1 and region 2, for the two
polarisation cases of TM and TE polarised light. . . . . . . . . . . . . . 130
8.7
The square magnitude of the Fourier coefficients for
n
= 0
... ±
4 with an
offset of δ = 0.42L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
8.8
The TE and TM reflectivity as a function of polar angle for an asymmetric
zigzag grating illuminated with a wavelength of λ
0
= 500 nm. . . . . . . 132
8.9
The dispersion of the SPP modes mapped as a function of TE and TM
reflectivity on an asymmetric zigzag grating. . . . . . . . . . . . . . . . . 133
8.10
The dispersion of the SPP modes mapped as a function of unpolarised
light reflectivity on an asymmetric zigzag grating. . . . . . . . . . . . . . 134
8.11
The iso-frequency contours at
λ
0
= 550 nm, measured on a symmetric
zigzag and an asymmetric zigzag grating. . . . . . . . . . . . . . . . . . 135
8.12
The calculated eigenmodes for the crossing of the
1
k
gx
and +2
k
gx
scattered SPPs at the first BZ for the cases of: a symmetric zigzag; a
symmetric, high-amplitude zigzag and; an asymmetric zigzag. . . . . . . 137
8.13
Field plots of
|E|
(colourplot) a for the low energy and high energy
standing waves at the first BZ. . . . . . . . . . . . . . . . . . . . . . . . 138
8.14
Field plots of
|E|
(colourplot) and
ˆ
E
(arrows) for the low energy and
high energy standing waves at the first BZ. . . . . . . . . . . . . . . . . 139
8.15
Experimental reflectivity colour plots mapping the dispersion of the band
gap around the 1st BZ. . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
8.16
The low energy and high energy
x
-component of the SPP standing wave
electric field and the incident field across two unit cells of the asymmetric
zigzag grating. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
8.17
Iso-frequency contours of an asymmetric zigzag grating for a range of
wavelengths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
8.18
The dispersion of the SPP modes on an asymmetric zigzag grating for
φ = 0
, and the scattergram for λ
0
= 450 nm. . . . . . . . . . . . . . . . 143
9.1
A possible experimental arrangement for the observation of collimated
surface waves using zigzag gratings. . . . . . . . . . . . . . . . . . . . . . 148
xiii
Chapter 1
Introduction
This thesis details original experimental investigations in to the interaction of light
with the mobile electrons at the surface of metallic diffraction gratings. The resulting
quantised surface waves, surface plasmon polaritons, have been investigated by optical
scientists for over a century, yet interest in the field has never been higher, and progress
has never been faster [
1
,
2
]. This is due in large part to the relatively recent advances in
nano-fabrication techniques, which have allowed greater control over the surface wave
propagation and dispersion characteristics on metallic surfaces. This work endeavours
to extend the understanding of surface plasmon polaritons on diffraction gratings by
exploring their propagation along symmetries and structures that have not been reported
previously, and whose fabrication has only recently become achievable.
1.1 History
The phenomenon of diffraction from gratings has been reported by scientists since
Francis Hopkinson peered at a street lamp through a silk handkerchief in 1785 [
3
,
4
].
The rainbow that is seen due to the dispersion of light through such fine structure was
recognised immediately as an incredibly useful optical property by Fraunhofer, and it
was not long before early diffraction gratings were being manufactured by scratching
fine grooves in to the surfaces of glasses and metals.
In 1902, it was these early forays into the structuring of metals that led to the first
recorded observations of surface plasmon polaritons’ interaction with light. Wood [
5
]
reported a bright band of enhanced reflection and a dark band of low reflection found
in the projected spectrum of a ruled speculum diffraction grating produced at John
Hopkins University [
6
]. Diffraction gratings produced by these early ruling engines were
finding a multitude of uses in physics at the time, particularly in the young field of
spectroscopy, and an explanation of Wood’s spectral anomalies was vigorously sought.
1
1. Introduction
Lord Rayleigh explained the observed dark band in his theory of diffraction gratings
in 1907 [
7
] as the wavelength of incident light at which a diffracted light ray will start,
or cease, to propagate. The redistribution of available propagating energy at this
point created a reflectivity feature corresponding to one of the two Wood’s anomalies.
Thereafter this was referred to as the ‘Rayleigh anomaly’ and today this is also called
the ‘diffraction edge’ or ‘pseudo-critical edge’.
Wood’s bright band was left unexplained for nearly thirty years until Strong [
8
],
attempting to improve the reflective efficiency of a set of gratings by coating them with
different types of metals, noticed the band shifted in energy depending on the metal
coating used. This dependence on the metal’s optical characteristics led Ugo Fano
[
9
11
], to the conclusion that this reflectivity anomaly was a signature of interactions
of light with a trapped surface wave, comprised in part by the metal’s conduction
electrons at the surface. He took the view that this represented a ‘zero-order’ waveguide
mode: a guided mode that could still exist localised to the surface when a hypothetical
supporting waveguide over-layer was decreased to zero thickness. Maxwell’s equations
provided Fano with a solution for such a wave, on the condition that one material was
a conductor and the other a dielectric
. Today we call this interface mode a surface
plasmon polariton (SPP) and it is defined as an electromagnetic surface wave bound to
the interface between a conductor and a dielectric.
While the first recorded observation of SPPs was on a diffraction grating, SPPs
are not unique to these devices alone, and are generally found in systems where a
conducting/dielectric interface exists. Once Fano had underpinned the physical origin
of this surface wave, and in particular the momentum requirements to resonantly drive
it, other methods by which to excite SPP on metal surfaces began to emerge. In the
late 1950s, Ritchie [
14
] and Ferrell [
15
] both predicted the excitation of SPPs on thin,
flat, metal films with accelerated electrons, with Ferrell also predicting the SPPs could
decay back into light. This prediction was confirmed experimentally by Steinmann [
16
]
two years later. Today, the use of electrons to excite SPPs has found application as a
very precise method of plasmonic investigation: electron energy loss spectroscopy [17].
In 1968, Otto [
18
] and simultaneously Kretschman & Raether [
19
] excited SPPs
using a prism geometry
. This technique used the evanescent ‘tail’ of light undergoing
total internal reflection in the prism to couple to SPPs on a metal/air interface, matching
the momentum of the SPP field through the classical tunnelling of the evanescent wave
originating from inside the high-index material. Their techniques for prism coupling are
Unbeknownst to Fano, similar work had been pursued by Zenneck [
12
] and also Sommerfeld [
13
] in
their research on the transmission of radio-waves across large distances using trapped surface waves
between seawater and air.
Turbadar [
20
] was the first to excite this wave using a prism geometry, but unfortunately failed to
connect his work to that of Fano or his observed phenomena to SPPs.
2
1. Introduction
still used today, particularly in SPP based sensors which can investigate a diverse range
of analytes, including the detection of fake tequila [21] and other spirits [22].
Research in to SPPs on diffraction gratings continued in parallel with the develop-
ment of these other excitation methods, with their observation and dispersion being
investigated in the 1960s [
23
25
]. Research in the 1970s and 1980s focussed on the
effect of deep grooves [
26
29
] and over-layer coatings [
30
,
31
], driven in part by the
development of holographic methods for diffraction grating manufacture [
32
]. This was
followed by work in the 1990s that explained the influence of the grating shape on the
resonances [
33
], SPP’s role in polarisation conversion [
34
] and the physical origin of
observed SPP band interactions [35].
With the advances in nano-fabrication techniques, the development of detailed
theoretical treatments for diffraction problems [
36
] and the increasing affordability
and power of computers, increasingly complex diffraction grating geometries could be
explored. The combinations of all these factors also helped spawn the active research
field of ‘plasmonics’, and further improvements in all these areas continue to be a driving
force behind the latest SPP research.
SPPs on gratings have already found many applications, and in some cases are
already integral to commercial products. With the energy of a SPP resonance being
highly sensitive to the local environment at the surface, SPPs have been used very
successfully in sensing applications [3739]; these have included the in-situ monitoring
of chemical reactions such as catalytic conversion [
40
,
41
], non-contact determination of
surface structure [42, 43], and the measurement of optical constants of metals [44].
Coupled SPPs on gratings have also been used to enhance various optoelectronic
devices; improvements to photo-detectors [
45
], the enhancement of laser beams [
46
49
]
and the ability to improve the efficiency of solar cells [
50
54
] have all been reported in the
literature. Generation of radiation has also been achieved on metallic or semiconducting
gratings through SPP mediated second harmonic generation [
55
] or by SPPs stimulating
electron photo-emission leading to the production of terahertz scale radiation [56, 57].
Additionally, SPPs on gratings have been employed as optical elements including
colour filters [
58
], polarisation converters [
59
], and various surface optics for collimation
or achieving negative refraction [
60
,
61
]. They have also been used for the novel demon-
stration of enhanced transmission through hole arrays [
62
65
] and the manipulation of
Magneto-optical effects such as Faraday and Kerr rotations [
66
]. The use of SPPs will
no doubt continue to find applications in the most recent of technologies, particularly
with very recent demonstration of their excitation on graphene [67].
This brief review of the investigations and applications of SPPs on gratings leads us
to the present day and the theme of this thesis. In this work, unconventional gratings
are used to couple to SPPs, and the propagation and dispersion of SPPs on such gratings
3
1. Introduction
(a) Rectangular (b) Oblique
(c) Symmetric Zigzag (d) Asymmetric Zigzag
Figure 1.1: The four types of grating geometry that are investigated in this thesis: (a)
The rectangular bigrating in chapter 5, (b) the oblique bigrating in chapter 6, (c) the
zigzag grating in chapter 7 and (d) the asymmetric zigzag grating in chapter 8.
is investigated. The experiments presented here are done to explore the properties of
SPPs on novel types of grating, and do not attempt to fit any particular application.
However, many possibilities for extensions or applications are possible, and these are
suggested in each chapter and summarised in the conclusions.
1.2 Scope and Outline of This Work
This thesis details experimental investigations into the propagation of SPPs on diffraction
gratings that possess novel structure or symmetries. Broadly it may be divided up in to
investigations of two types of diffraction grating; ‘crossed’ bigratings and a new type
of diffractive optical element: the ‘zigzag grating’. Since both these types of grating
possess two different diffractive periods in their surface geometry, they may both be
considered a type of the larger family of metallic ‘bigratings’. There are four diffraction
grating geometries investigated in this work, and these are summarised in figure 1.1.
They are; the rectangular bigrating, the oblique bigrating, the ‘zigzag grating’ and the
‘asymmetric zigzag grating’.
The background theory of SPPs on both planar films and on metallic diffraction
gratings is presented in chapter 2. This chapter covers the origin, coupling conditions
4
1. Introduction
and band-structure of SPPs on both planar and periodic surfaces. The methods by
which the optical response of the gratings under consideration have been calculated
theoretically are then explained in chapter 3.
Chapter 4 details the experimental methods used for the production and measure-
ments of gratings in this thesis. In addition to the standard experimental techniques
used, a new method by which to map the plasmonic analogue to the iso-frequency
‘Fermi-contours’ of the SPP band structure is presented. These iso-frequency contours
have been recorded using imaging scatterometry and this new technique, developed as
part of this body of work, is used extensively throughout this thesis.
Chapter 5 presents some experimental observations on the excited SPPs supported by
rectangular bigratings. These are gratings formed of two diffraction gratings of different
pitch, crossed relative to each other at an angle of 90
. The dispersion of these modes
on the surface and the formation of standing surface-wave states are experimentally
recorded and matched to a theoretical model. It is found that by deepening the grooves
of one of the constituent diffraction gratings, the propagation of SPPs along the surface
becomes highly anisotropic. Control over this effective mode index in different directions
along the surface could find application in surface-wave optics devices.
The work on ‘crossed’ bigratings continues in chapter 6, with the experimental
investigation of SPPs on a bigrating with the reduced symmetry of an oblique lattice.
The dispersion and scattering mechanisms on a fabricated oblique grating are recorded
experimentally and explained. SPP mediated polarisation conversion is also observed
on these gratings, as the scattered surface fields propagate along a surface of no mirror
symmetry. The lack of symmetry on such a grating leads to the observation of SPP
band gaps not forming at the Brillioun Zone (BZ) boundaries, and a general discussion
of the oblique symmetry constraints on SPPs is offered to explain why this is so.
Chapter 7 introduces a new type of diffraction grating, the ‘zigzag grating’. This
grating uses sub-wavelength structure in one direction to introduce an diffractive
periodicity in the orthogonal direction. It is found, experientially and theoretically, that
even-order diffracted fields only couple to SPPs for one linear polarisation of light, while
odd-order diffracted fields only couple to the other, orthogonally polarised light. Further,
it is shown that SPP band gaps are forbidden at the first BZ boundary by the symmetry
of the zigzag surface. Finally in this chapter, it is shown that the sub-wavelength grooves
on such a grating can lead to highly anisotropic SPP propagation. This anisotropy leads
to SPP propagation at certain frequencies in only one single direction, irrespective of
excitation angle. When combined with the lack of band gaps on such a symmetry, this
makes these gratings excellent candidates for surface wave collimation devices.
The final experimental results of the thesis are presented in chapter 8, which extends
the zigzag grating geometry to one with reduced symmetry. This relaxes the polarisation
5
1. Introduction
conditions of the previous zigzag grating, leading to any SPP being coupled to with
any incident polarisation of light, a result which could prove relevant to improving
the efficiency of many plasmonic devices. The band structure of the SPPs supported
by this grating is also experimentally investigated and the coupling to the different
standing wave states by light which occur at the first BZ boundary is explored. The SPP
anisotropy previously found for a zigzag grating is also found in this new asymmetric
zigzag grating case, and combined with the large band gaps which form at the first BZ
boundary, the grating is shown to support a full plasmonic band gap, for which SPP
propagation is forbidden in all directions.
The thesis is concluded in chapter 9 with a summary of the findings and suggested
future research which could extend the findings and applications of this work.
6
Chapter 2
Background Theory
2.1 Introduction
The interaction of light with metals has long proven to be a reliable path to striking
optical effects. The Lycurgus cup is an example of a 4th century plasmonic device;
a glass Roman goblet which uses colloidal gold and silver nano-particles to achieve a
dull green surface reflection, but glows blood red when illuminated from the inside
[68]. Resonant interaction between light and the electrons in these suspended metallic
nano-particles provide the mechanism for this vivid effect, the same effect found in the
majority of stained glass windows.
The topic of this thesis also concerns the coupling of light to electrons, specifically
the mobile electrons found at the interface between dielectric and a conductor. Surface
plasmon polaritons are electromagnetic surface waves coupled strongly to the longitudinal
oscillations of this free electron plasma at the interface. They are quantised surface
waves that propagate along the interface, evanescently decaying in the normal direction.
This chapter introduces the background electromagnetic theory of these surface waves
and their interaction with light.
Sections 2.2 and 2.3 deal with the origin and characteristics of surface plasmon
polaritons propagating along a flat interface. Metallic diffraction gratings provide a
mechanism by which to couple light to these surface waves. This coupling, and the
constraints placed upon surface waves on a such periodic surfaces, are detailed in section
2.4.
2.2 Surface Polaritons on Planar Surfaces
Surface plasmon polaritons (SPPs) can be categorized as a member of a larger family of
surface waves, broadly named ‘surface polaritons’. Surface polaritons are electromagnetic
7
2. Background Theory
surface waves coupled strongly to an elementary excitation and bound evanescently to
the interface between two media. When a photon couples to the longitudinal oscillations
of the free-electron plasma on metal surfaces, the resulting surface polariton is called a
surface plasmon polariton. Surface polaritons also couple with surface lattice vibrations
(surface phonon polaritons) [
69
], surface electron-hole pairs in semiconductors (surface
exciton polaritons) [
70
] and collective excitations of surface electron spin (surface magnon
polaritons) [
71
]. More recently, the manufacture of resonant sub-wavelength structures,
or ‘metamaterials’, has allowed the design of surfaces with tailored resonances other
than such elementary excitations, which may couple strongly to photons and produce
surface polaritons [
72
,
73
]. The resulting surface polaritons are often referred to as a
‘spoof surface plasmon polaritons.
Solving Maxwell’s equations for an evanescently bound electromagnetic wave at an
interface leads to a similar relationship between energy and momentum for all surface
polaritons. The physical origin of the different types of surface excitations and their
coupling to light is expressed in terms of the dielectric and magnetic functions which
are frequency dependent and complex in general.
Fano [
10
] derived the dispersion relation for a trapped surface wave by first consider-
ing light propagating in a glass plate of finite thickness, bounded by semi-infinite vacuum
and metal half-spaces. In a simple waveguide such as this, total internal reflections
prevent the propagating light from escaping into the bounding medium. The necessary
continuity of electric field across a boundary for light undergoing total internal reflection
also requires non-propagating ‘evanescent’ waves extending into the bounding media,
which do not transfer any power. The question as posed by Fano [
10
] was then: ‘Is
there left any proper value when the thickness vanishes?’. As the thickness of the glass
plate tends to zero, there is indeed still a valid solution to Maxwell’s equations for a
wave travelling along the surface, evanescently bound in the normal direction; a surface
polariton. In these terms, a surface polariton can be thought of as the lowest-order
waveguide mode.
The dispersion of the surface polariton considered in this thesis, the surface plasmon
polariton on a metal/dielectric interface, allows us to simplify the problem to that of
isotropic, non-magnetic media. The derivation of this relationship [
74
] is presented
below, and is also valid for other surface polaritons in isotropic non-magnetic media,
such as surface phonon polaritons.
A schematic diagram of the system is shown in figure 2.1. Light illuminates a
planar surface between two media of permittivities
ε
1
and
ε
2
in the
xz
plane, which we
define to be the plane of incidence. The wavevector of the light in the
m
th
medium is
although the inclusion of ‘plasmon’ is not strictly correct as the photon is not coupled to the free
electron plasma oscillations but more usually a sub-wavelength cavity resonance.
8
2. Background Theory
E
0
E
1
E
2
ˆx
ˆz
ε
2
ε
1
k
0
k
1
k
2
Figure 2.1: A schematic representation of propagating electromagnetic fields at an
interface between two materials. The green rays illustrate the direction of the waves,
which are coincident with the wavevectors,
k
. The electric field is polarised in the
xz
plane, represented by black arrows. The optical response of the two media are
characterised by their permativities,
ε
1
and
ε
2
. The
ˆy
-direction is out of the page and
the position z = 0 is at the interface.
k
m
=
k
x
m
ˆx
+
k
z
m
ˆz
[
k
x
m
,
0
, k
z
m
], with no component in the
ˆy
-direction (out of the
page). The light is then reflected back into medium 1 and refracted into medium 2.
For light polarised with the electric field parallel to the plane of incidence (Transverse
Magnetic, or TM polarised) the plane waves can be described as,
E
m
= [E
x
m
, 0, E
z
m
] e
i(k
m
·x+k
m
·z)
e
t
H
m
= [0, H
y
m
, 0] e
i(k
m
·x+k
m
·z)
e
t
where
ω
is the angular frequency of the light,
t
is time and
m
is a subscript indicating
in which medium the field is propagating.
H
y
is the component of magnetic field in
the
ˆy
-direction and
E
x,z
is the component of electric field in the
ˆx
and
ˆz
directions,
respectively. Since the surface polariton is a trapped surface wave, we set the incident
wave to zero. Setting
E
0
to zero, we are left with two sets of fields (electric and magnetic)
for the half-spaces above and below the interface,
z > 0
E
1
= [E
x
1
, 0, E
z
1
] e
i(k
x
1
x+k
z
1
z)
e
t
H
1
= [0, H
y
1
, 0] e
i(k
x
1
x+k
z
1
z)
e
t
(2.1)
z < 0
E
2
= [E
x
2
, 0, E
z
2
] e
i(k
x
2
xk
z
2
z)
e
t
H
2
= [0, H
y
2
, 0] e
i(k
x
2
xk
z
2
z)
e
t
(2.2)
We may combine the equations for electric and magnetic field in the two half-spaces
9
2. Background Theory
using Amp`ere’s law,
× H
m
= ε
m
E
m
t
where
ε
m
is the permittivity in medium
m
. The relationships between the tangential
electric and transverse magnetic fields in each material are then,
k
z
1
H
y
1
= +ωε
1
E
x
1
(2.3)
k
z
2
H
y
2
= ωε
2
E
x
2
(2.4)
Having obtained expressions for the electromagnetic fields in both media, we must now
consider the continuity of these fields over the boundary. At the interface the boundary
conditions for the electromagnetic waves are expressed as,
E
x
1
= E
x
2
(2.5)
H
y
1
= H
y
2
(2.6)
ε
1
E
z
1
= ε
2
E
z
2
(2.7)
These are the conditions that tangential electric fields, transverse magnetic fields and
the normal component of the electric displacement vector (
D
z
m
=
ε
m
E
z
m
) must be
continuous across the boundary at
z
= 0. The continuity of the tangential electric field
(equation 2.5) allows us to combine equations 2.3 and 2.4,
k
z
1
ε
1
H
y
1
+
k
z
2
ε
2
H
y
2
= 0
and continuity of transverse magnetic field (equation 2.6) then yields
k
z
1
ε
1
+
k
z
2
ε
2
= 0 (2.8)
Finally, to obtain a relationship in terms of the momentum of the surface wave in the
ˆx
-
direction (
k
x
), we consider the conservation of momentum in both regions. Conservation
of tangential momentum requires that
k
x
1
=
k
x
2
=
k
x
and so the expression for total
conserved momentum is,
k
2
x
+ k
2
z
m
= ε
m
ω
c
2
(2.9)
Obtaining expressions for
k
z
1
and
k
z
2
using this conservation of momentum expression
and combining them with equation 2.8, we find the dispersion relation for the surface
At this point, we also employ some mathematical sleight of hand to substitute the permittivity,
ε
m
, for the relative permittivity (
ε
m
0
) by cancelling the common factor of
ε
0
in
ε
1
and
ε
2
. For clarity
we redefine ε
m
as the relative permittivity for the remainder of this thesis.
10
2. Background Theory
polariton,
k
x
=
ω
c
r
ε
1
ε
2
ε
1
+ ε
2
(2.10)
This equation relates the angular frequency of the field,
ω
to the wavevector along
the surface,
k
x
. The energy and momentum of the surface mode is related to these
quantities respectively by a factor of the reduced Planck’s constant,
~
. Because of this,
it is also often referred to as the energy-momentum relation.
The final observation to be made in regards to the surface polariton dispersion is
the set of
ε
1
and
ε
2
values which can support a bound surface wave. The continuity
of normal electric displacement was mentioned briefly in equation 2.7, and we shall
re-print it here.
ε
1
E
z
1
= ε
2
E
z
2
This boundary condition states that the normal electric displacement must remain
continuous across the interface. However, from the plane wave equation sets (equations
2.12.2) it is apparent that
E
z
1
is always 180
out-of-phase with
E
z
2
.
E
z
1
will always
be of opposite sign to E
z
2
. This is intuitive, as induced surface charge at the interface
will naturally cause the electric field to extend into each surrounding media in opposite
directions. To satisfy the boundary condition, and so support surface polaritons on such
a surface, one material must be capable of ‘inverting’ the electric displacement resulting
in the condition that the permitivities ε
1
and ε
2
must also be of opposite signs.
2.3 Dispersion of SPPs on Planar Surfaces
The optical response of a metal is characterized by the metal’s frequency dependent
permittivity, also called the ‘dielectric function’ of the material. This optical response
function is dominated by two characteristics of metals; the fact that conduction electrons
are free to move within the bulk of the material, and the presence of inter-band transitions
of the valence electrons in the atomic orbitals. In the visible region, conduction electron
behaviour is ballistic, oscillating and interacting with light many times before scattering
from the crystal or other electrons. The inter-band transitions for a noble metal such
as silver occur at the edges of the visible spectrum and so the dominant effect in the
appearance of silver is due mostly to the strong interaction of the free conduction
electrons with optical fields. At the surface of the metal, electromagnetic fields coupled
to longitudinal oscillations of this free-electron plasma form together a surface polariton:
the surface plasmon polariton.
The dispersion of a surface plasmon polariton takes the same mathematical form as
equation 2.10 with one dielectric function representing the frequency dependent metal,
11
2. Background Theory
ε
2
(
ω
), and the other representing a bounding non-conducting crystal with a dielectric
constant, ε
1
,
k
x
=
ω
c
s
ε
1
ε
2
(ω)
ε
1
+ ε
2
(ω)
(2.11)
A useful approximation for the metal dielectric function was presented by Drude
[
75
]. Drude considered free electrons travelling in a classical manner, scattering from the
ionic lattice with a characteristic average scattering rate. Electron-electron scattering is
disregarded and the permittivity is expressed as,
ε
2
(ω) = ε
ω
2
p
ω
2
+ γ
(2.12)
where
ω
is the angular frequency of the light,
γ
is the average rate of collision of the
free conduction electrons with the lattice, and
ε
is the response of the ionic lattice,
which is frequency independent and for metals in the visible domain equals 1. The
‘plasma frequency’ of the metal,
ω
p
, is the natural frequency of oscillation of the bulk
conduction electrons.
An example of the permittivity of silver using the Drude model is show in figure 2.2.
It shows, for all visible regime wavelengths, that the real part of the silver permittivity
is negative and there is a non-zero imaginary component. This is generally true for a
wide range of metals at visible wavelengths. This negative permittivity of metals in the
visible domain, when bounded by a dielectric with a positive real permittivity, satisfies
the boundary conditions (equation 2.7) for a surface plasmon polariton. Substituting
2.12 in to the dispersion relation for a surface plasmon polariton (equation 2.11), we
obtain a fair approximation of the dispersion relation for a surface plasmon polariton on
a planar metal film. This is plotted in figure 2.3. Figure 2.3 shows the dispersion of a
surface plasmon polariton on a flat interface between silver and air. Plotted on the same
scale is a line representing a grazing photon along the surface, the ‘light-line’. Notice
that the light line and the surface plasmon polariton dispersion line do not cross. There
is no solution at which the energy and momentum of free-space light is equal to that
of the surface plasmon polariton. Since for a given energy of light, the light possesses
insufficient momentum to match that of an SPP, the conclusion is that free-space light
incident on a flat metallic surface cannot resonantly drive a surface plasmon polariton.
2.3.1 Penetration Depth
An excited SPP at the interface between a conductor and dielectric will possess elec-
tromagnetic fields which decay exponentially into both bounding media. A useful
measure of this decay is the penetration depth,
L
z
, which is the distance at which the
12
2. Background Theory
400 500 600 700 800 900
-35 -30 -25 -20 -15 -10
0.5 1 1.5 2 2.5 3
wavelength (nm)
ε
r
ε
i
18.6
0.92
Figure 2.2: The Drude model calculated with
ω
p
= 1
.
32
×
10
16
Hz, γ
= 1
.
4
×
10
14
Hz
.
As an example, the wavelength for a HeNe laser (632.8 nm) is highlighted, showing the
permittivity of silver at this wavelength to be ε(632.8 nm) 18.6 + 0.92i
.
field amplitude has decreased to 1
/e
of it’s maximum value. Momentum conservation
for a SPP (where
k
x
> ε
m
k
0
) gives an expression
k
z
m
=
p
ε
m
k
2
0
k
2
x
(equation 2.9
simplified.), which leads to the conclusion that
k
z
m
for SPPs must be purely imaginary.
Substituting an imaginary
k
z
m
into an expression for the electric field at the surface
gives
,
E
m
= [E
x
m
, 0, E
z
m
] e
ik
x
x
e
k
z
m
z
(2.13)
which is an expression for an electric field which does indeed decay exponentially into
the two bounding media, and travels along the surface in the
x
direction. The value of
z
for which
E
z
m
falls to
e
1
of the maximum value is then 1
/k
z
m
. Substituting equation
2.10 into equation 2.9 we find that the expression for k
z
m
is given by,
k
z
m
= ±k
0
s
ε
m
ε
1
ε
2
ε
1
+ ε
2
= ±k
0
s
ε
2
m
ε
1
+ ε
2
(2.14)
the time dependent term, e
iωt
has been omitted for clarity.
13
2. Background Theory
in-plane wavevector, k
x
(m
1
)
angular frequency, ω (rad s
1
)
0 2×10
7
4×10
7
6×10
7
8×10
7
0 2×10
15
6×10
15
10×10
15
ω
p
2
Figure 2.3: The dispersion of a surface plasmon polariton on a planar film approximated
with the Drude model with ω
p
= 1.32 × 10
16
Hz, γ = 1.4 × 10
14
Hz.
We may simplify this expression for the case under consideration with medium
m
= 1
being a non-absorbing dielectric,
Re
(
ε
1
)
>
0 and
Im
(
ε
1
) = 0, and medium
m
= 2 a lossy
metal,
Re
(
ε
2
)
<
0 and
Im
(
ε
2
)
>
0. In the case where the metal is highly conducting
we also have the considerations
|Re
(
ε
2
)
|
1 and
|Re
(
ε
2
)
| Im
(
ε
2
). Under these
conditions, equation 2.14 simplifies to,
k
z
m
= ±k
0
s
Re(ε
m
)
2
Re(ε
2
)
(2.15)
The penetration depth is then,
L
z
m
=
1
k
z
m
= λ
0
1
2π
s
|Re(ε
2
)|
|Re(ε
m
)
2
|
(2.16)
For a typical planar silver surface in air with
k
0
= 2
π
0
= 2
π/
632
.
8nm, the penetration
depth for the SPP into the air is calculated, using equation 2.16, to be
L
z
1
415 nm and
in to the metal
L
z
2
24 nm. The field at the surface and this associated exponential
14
2. Background Theory
ε
2
ε
1
E
++ ++−−
(a)
ε
2
ε
1
|E
z
|
z
(b)
Figure 2.4: (a) Diagram of the electric field vectors of a SPP at the interface between a
metal and a dielectric. (b) The exponential decay of the
E
z
component away from the
surface, with a maximum amplitude at
z
= 0 and the different penetration lengths for
the two materials shown.
decay of the SPP fields in the
z
direction is shown diagrammatically in figure 2.4 for an
SPP in the visible regime.
In the limit of a perfect conductor, where
ε
2
−∞
, equation 2.16 shows that the
penetration depth into the metal
L
z
1
0 and the penetration depth into the dielectric
tends to
L
z
2
. With no field in the metal and the ‘decay length’ in
z
never actually
decaying in the dielectric medium, the result is a grazing plane wave and no localisation
of the field to the surface can be said to exist. This is why the excitation of SPP
waves in the microwave regime is not observed on unaltered planar surfaces, as at these
frequencies the metals approximate a perfect conductor and so the vanishingly-small
localisation of the bound surface modes leads to the SPP resembling a grazing photon.
2.3.2 Propagation Length
By similar considerations of the surface electric field, we define the propagation length
of the SPP along the surface as the length at which the field intensity has fallen to
e
1
of its maximum value. While the SPP propagates in the
x
direction, it will be damped
by Joule losses to the metal. This leads to the expression for
k
x
to be generally complex,
and the imaginary component of of
k
x
again provides the exponential decay of the SPP
as it travels along the interface. The imaginary part of k
x
is given by,
Im(k
x
) =
k
0
Im(ε
2
)
2 Re(ε
2
)
2
ε
1
Re(ε
2
)
ε
1
+ Re(ε
2
)
3
2
(2.17)
15
2. Background Theory
This gives the propagation length, L
x
as,
L
x
= λ
0
Re(ε
2
)
2
2π Im(ε
2
)
ε
1
+ Re(ε
2
)
ε
1
Re(ε
2
)
3
2
(2.18)
For a typical planar silver surface in air with
k
0
= 2
π
0
= 2
π/
632
.
8nm, the propagation
length is
L
x
= 42
.
6
µ
m. This propagation length is large enough so the SPP intensity is
sufficient to interact strongly and Bragg scatter on diffraction gratings with sub-micron
periodicity, a topic we shall cover in the next section.
2.4 Surface Plasmon Polaritons on Diffraction Gratings
2.4.1 Diffraction Gratings
A diffraction grating is an optical device in which the dielectric constant varies periodi-
cally across the surface, and whose period is of the order of the wavelength of light. This
periodic variation leads to the diffraction of light by way of localised phase changes in the
impinging field, resulting in either the coherent constructive or destructive interference
of the waves in the far-field.
The first recorded observation of diffraction by fine structure was by Francis Hopkin-
son in the late 1700s. Peering at a street lamp through a silk handkerchief, Hopkinson
observed rainbows and enquired with his friend, Rittenhause as to the cause. Ritten-
hause then went on to manufacture and study the first reported ‘diffraction grating’
made of parallel hairs, set between two brass wires cut with a fine screw head [
3
]. This
was the first example of a transmission grating and the device’s dispersive nature.
In this thesis, a periodic modulation across a reflecting surface is achieved by surface-
relief. Shallow grooves are cut into a metal surface with a defined period, on the order
of the wavelength of visible light, causing diffraction in the reflected light from the
surface. Other methods by which diffraction gratings may be produced exist, such as
phase-modulated diffraction gratings [
32
] or periodic hole arrays, but these will not be
dealt with here.
The coordinate system for a simple reflection grating of this type is shown in figure
2.5. The grating consists of a set of periodic grooves in a metal surface, with the
amplitude varying along the
x
-axis, with a periodicity of
λ
gx
. Light impinges on the
grating surface at a polar angle
θ
, in a plane of incidence defined at an azimuthal angle
φ
, where
φ
= 0
corresponds to when the plane of incidence is perpendicular to the
grating grooves. The grooves are a depth
d
. The polarisation of the incident light
is defined with respect to the plane of incidence, such that Transverse Electric (TE)
polarised light is when the electric field,
E
T E
is oriented perpendicular (transverse) to
16
2. Background Theory
φ
θ
E
T M
E
T E
d
x
y
z
λ
gx
plane of incidence
Figure 2.5: The coordinate system used for a simple grating in the conical mount.
the plane of incidence, and Transverse Magnetic polarised light is defined as the electric
vector E
T M
lying within the plane of incidence.
In this orientation, where
φ
is allowed to vary and is not necessarily equal to 0
, the
diffracted beams form in general a ‘cone’ of diffraction, and so it is named the ‘conical’
mounting/arrangement.
The relationship between incident and diffracted light in the conical mount is given
by the equation,
n
λ
0
λ
gx
= (sin θ cos φ + sin θ
n
cos φ
n
) (2.19)
where
n
is an integer denoting the spectral order of diffraction,
λ
0
is the incident
wavelength and
θ
n
, φ
n
is the polar and azimuthal angle of the
n
th
diffracted order,
respectively.
The wavevector of light is defined as
k
0
= 2
π
ˆ
r
0
, where
ˆ
r
is the unit vector denoting
the direction of the wave. This wavevector is associated with the light’s momentum by
a factor of the reduced Planck’s constant, such that
p
=
~ k
0
, and it’s scalar equivalent
is the wavenumber (
k
0
). Similarly, the grating wavevector is defined, in this case, as
k
gx
= 2
π
ˆ
x
gx
(its wavenumber is
k
gx
). We may re-write equation 2.19 in terms of
17
2. Background Theory
these quantities,
k
0
sin θ
n
cos φ
n
= k
0
sin θ cos φ nk
gx
(2.20)
k
k
n
= k
k
0
nk
gx
(2.21)
where
k
k
n
is the wavevector component of the diffracted light parallel to the surface and
k
k
0
is the incident light’s wavevector parallel to the surface. So a diffraction grating
modifies the surface wavevector of the incident light by the addition or subtraction
(since n may be positive or negative) of an integer number of grating wavevectors.
Recall from section 2.3 that the coupling of free-space light to surface plasmon
polaritons is not possible on flat surfaces due to the mismatch between the SPP and
light’s wavevectors (or, equivalently, the momentum mismatch). Since the diffraction
grating allows us to modify the wavevector of incident light, the diffraction grating may
be used to couple to these previously unmatchable modes. This will be explored in the
next subsection.
2.4.2 Coupling Surface Plasmons to Light by Coherent Scattering
The three illustrations in figure 2.6 show the momentum-space diagrams for monochro-
matic light of frequency
ω
0
(=
ck
0
) incident on three gratings of different pitch, at a
fixed polar angle in the classical mount of
φ
= 0
. The allowed real momentum states
of propagating light are represented as a red circle of radius
k
0
. Conservation of energy
requires that any vectors representing propagating light have a magnitude of
k
0
, and
as such are represented as radial arrows of this circle. Points that lie outside of the
red light circle possess greater momentum than the incident light circle and represent
non-propagating (evanescent) light with imaginary momenta. The red stars on the
diagram are the allowed momentum states of a surface plasmon polariton. The surface
plasmon polariton momentum state lies at a finite real value of
k
x
, with no real value of
k
z
, indicating the surface mode has a purely imaginary value of
k
z
and so evanesently
decays normal to the surface while propagating along it. Notice the surface plasmon
polariton state lies outside the light circle, illustrating that even grazing light possesses
insufficient momentum to match that of the surface mode.
The in-plane (
k
x
) component of light momentum on a grating surface is altered by the
addition of an integer number of grating vectors (equation 2.21). In figure 2.6(a), three
possible scattering events of incident light are shown, a +1
k
g
, a 1
k
g
and no scattering
(0
k
g
). The allowed momentum states of light with these new values of
k
x
show that
For the conical mount, these allowed momentum states would form instead a hemisphere in
three-dimensional k-space.
18
2. Background Theory
k
x
k
z
k
0
air
metal
k
SP P
k
SP P
k
x
+ k
g
k
x
k
g
(a)
k
x
k
z
k
0
air
metal
k
SP P
k
SP P
k
x
+ k
g
k
x
k
g
(b)
k
x
k
z
k
0
air
metal
k
SP P
k
SP P
k
x
+ k
g
k
x
k
g
(c)
Figure 2.6: The allowed real momentum states for light incident on three diffraction
gratings. The gratings reduce in period (increase in
k
g
) from (a) to (c). Points along
the red circle are the possible momentum states for light in air (
k
0
). There is no
corressponding circle in the metal halfspace as light in this medium is considered to be
evanescently damped and so has no real value of momentum. The red stars show the
momentum states for SPPs.
19
2. Background Theory
diffracted light must leave the surface at a different polar angle to the specular reflection
(0
k
g
) to conserve total momentum,
k
0
(and so still lie on the radial line of constant
total momentum). This angle of diffraction is dependent on the size of the light circle,
determined by the frequency of the radiation, which illustrates the dispersive properties
of a diffraction grating. The grating represented in figure 2.6(b) has a periodicity less
than that of grating 2.6(a). The period has been chosen to illustrate the case where
the +1
k
g
diffracted order is grazing the surface. An increase in the polar angle, or a
decrease in the incident light wavelength would leave no available propagating light
momentum states for the +1
k
g
order. The order would become evanescent. To satisfy
energy conservation, this transition from propagating diffracted order to evanescent
light requires the power of that order to be re-distributed to the remaining propagating
orders. In spectra or angular data this is shown as a step in the intensity of reflected
light. First observed by Wood [
5
] and explained by Rayleigh [
7
], this feature is referred
to in literature as a diffraction edge, a critical edge or a Rayleigh anomaly. In this thesis
we will adopt the convention of calling this feature a diffraction edge.
Finally, figure 2.6(c) shows a grating of sufficient pitch where a +
k
g
scattering
photon has sufficient momentum to match the momentum state of a surface plasmon
polariton. This light’s value of in-plane momentum is
k
x
+
k
g
. The grating has coupled
free space light to the surface mode by enhancing the in-plane wavevector of the light
to match to that of the surface plasmon polariton. Momentum conservation is still
maintained as, while the real part of
k
z
= 0, the imaginary part is not. Using equation
2.19 this condition is expressed in this case as,
k
SP P
= k
1
= k
x
+ k
gx
(2.22)
Notice that this scattered light vector is now greater than the radial
k
0
circle, showing it
may no longer match a momentum state of propagating light and so does not propagate
away from the surface. The light is an evanescent diffracted order which resonantly
drives the surface plasmon polariton. This case may be represented on the dispersion
diagram for a SPP as shown in figure 2.7(a), where the zero-order SPP dispersion now
passes through the diffracted light cones (in blue). For all the points along the SPP
dispersion which lie inside the diffracted light cones (represented as blue circles along
the SPP dispersion (black line)), coupling can occur between the evanescent light and
the SPPs. Since all of this occurs in the non-radiative region, no optical effect related
to the SPP excitation is observed in this case. However, this surface plasmon polariton
may also decay back into the zero-order reflectivity by itself diffracting by
1
k
g
. The
light returning to the specular order, having undergone two scattering events (and an
excitation and decay event), will be out of phase with the incident light leading to a
20
2. Background Theory
(a)
(b)
Figure 2.7: Two dispersion diagrams illustrating in-plane diffractive coupling to the (a)
zero-order SPP in the extended zone scheme and (b) the higher-order diffracted SPPs
couling to zero-order light in the reduced zone scheme.
sharp drop in the reflectivity observed. The observation of this interaction on the SPP
dispersion is shown in figure 2.7(b). Light diffracts and couples to the zero-order SPP,
then diffracts and decays back into propagating light inside the zero-order light cone.
For the SPP dispersion points lying inside the (red) light cone, we expect there to be an
effect on light matching these momentum and energy values. It is exactly equivalent to
interpret figure 2.7(b) as the zero-order SPP curve (black) itself diffracting into the light
cone, as the surface waves themselves Bragg scatter to interact with zero-order light.
Typically this interpretation will be favoured in this thesis, as it allows a simplified
discussion for mode interaction without sacrificing any scientific rigour.
This coherent scattering of SPPs will produce extrema in the specular reflectivity of
the grating indicative of the light having interacted with a SPP. By mapping the zero-
order reflectivity of a diffraction grating as a function of
ω
and
k
, we may reproduce the
dispersion curves of the SPPs, with the reflectivity extrema occurring at the diffracted
SPP mode positions.
The shape of these dispersive bands is dependent on the plane of incidence, the
orientation of the grating and, in the case of bigratings, covered later, the additional
available scattering grating vectors. Figure 2.8 shows three possible intersections of
a plane of incidence with scattered SPP cones in (
ω, k
x
, k
y
) space and their possible
coupling. Since the SPP dispersion is approximated to be isotropic in all directions
,
This is not generally true on grating surfaces, as the grooves have destroyed the isotropy of the
21
2. Background Theory
k
y
k
x
ω
(a)
k
y
k
x
ω
(b)
k
x
k
y
ω
(c)
Figure 2.8: Intersections of the plane of incidence (
k
y
= 0) for (a) the unscattered
surface plasmon polariton cone (b) a surface plasmon polariton cone diffracted in the
plane of incidence and (c) a surface plasmon polariton cone diffracted perpendicular
to the plane of incidence. The projected intersection screen also shows the radiative
(white) and non-radiative (grey) regions of the light cone.
22
2. Background Theory
the SPP dispersion can be shown as a cone formed by sweeping the dispersion line
around the
ω
axis. Figure 2.8(a) is the non-scattered zero-order SPP cone. The plane
of incidence and the conic-like intersection(along
k
x
in all cases here), is projected in a
back plane, with the region of propagating light shown in white, and the evanescent
light in grey. In this case, the experiment would yield the same dispersion as shown in
figure 2.3, with none of the available (blue) SPP momentum states passing through the
region of free-space light. In figure 2.8(b), a SPP cone has been scattered in the plane
of incidence by 1
k
g
. The projected intersection in the plane-of incidence shows that now
a portion of the SPP momentum states lies within the white region of free-space light,
and so observable coupling can occur. The shape of the SPP band is the same as the
in-plane scattered SPPs shown in figure 2.3.
Finally, figure 2.8(c) shows the results of a SPP cones having scattered in a direction
90
to the plane of incidence. This can occur for a monograting oriented so that
φ
= 90
,
or this could be the result of the SPP undergoing a different possible scattering event
on a square bigrating. The SPP momentum of this scattered mode does cross the region
of free space light (white area on the projected screen), and forms a flatter band shaped
as shown which may couple to the free-space light. These figures 2.8(b) and 2.8(c) show
two possible functional forms of coupled SPPs in the plane of incidence. Real diffraction
gratings will contain many of these scattering mechanisms, and the mapped dispersion
will consequently be formed of multiple versions of these SPP bands.
2.4.3 Surface Plasmon Polaritons on Bigratings
Bigratings are defined as diffraction gratings which possess two grating vectors which do
not lie collinear to each other. The grating equation from equation 2.21 then becomes,
k
k
(n
x
,n
v
)
= k
k
0
± n
x
k
gx
± n
v
k
gv
(2.23)
where the variables are defined as they were for equation 2.21, with
k
gv
the grating
vector of the second pitch which is oriented at an angle
α
to the
x
-direction.
n
x
is the
integer number representing the spectral order in the
x
direction and
n
v
is the spectral
order in the
v
direction. In this thesis we shall occasionally use the notation (
n
x
, n
v
)
to refer to different diffracted orders, for example a (+1
,
+1) scattered SPP would be
represented as k
SP P
= k
k
0
+ 1k
gx
+ 1k
gv
.
With this simple definition, all the possible 2D Bravais lattices for a diffraction
grating can be realised. With the angle
α
equal to the angle between the gratings in
real space (the corresponding angle in reciprocal space is
α
?
= 180
α
), we have the
following definitions listed in table 2.1.
surface on which the SPPs propagate.
23
2. Background Theory
Bravais Lattice α k
gx
, k
gv
References
Square α = 90
|k
gx
| = |k
gv
| [44, 52, 53, 7678]
Rectangular α = 90
|k
gx
| 6= |k
gv
| [7981][This Work]
Hexagonal α = 60
|k
gx
| = |k
gv
| [44, 58, 76, 82]
Oblique α 6= 90
|k
gx
| 6= |k
gv
| [This Work]
Rhombic α 6= 90
|k
gx
| 6= |k
gv
| [83]
Table 2.1: The Bravais lattice types and associated work on SPPs on such symmetries.
Rhombic is also called centred-rectangular, see Kittel [75] for an example.
k
gv
k
gx
Figure 2.9: The reciprocal space map for an oblique bigrating with grating vectors
k
gx
and
k
gv
. The black lines represent zero-order modes, the blue lines represent diffracted
modes. Lines represent grazing photon momentum states, dashed lines represent SPPs.
The coupling of light to SPPs on such a bigrating is best visualised in the
k
-space
diagrams for SPPs on such lattices. Figure 2.9 shows an oblique lattice in
k
-space with
α
= 60
. The two grating vectors,
k
gx
and
k
gv
are illustrated as red arrows. The black
solid circle represents the momentum of a grazing zero-order photon and so the inside
of this circle contains all the possible momentum states of propagating free-space light.
The momentum states of this diffracted light are shown as blue circles, centred around
24
2. Background Theory
400 500 600 700 800
0.0 0.2 0.4 0.6 0.8 1.0
wavelength (nm)
R
T M
+1+2
diffraction edge
+1 SPP
inter-band transition absorptions
diffraction edge (weak)
+2 SPP
Figure 2.10: An example spectrum for a simple sinusoidal silver grating, calculated using
the Chandezon method. the grating has the following parameters:
λ
gx
= 750 nm, and
depth of 40 nm with a frequency dependent dielectric function for silver from literature
[
84
]. The spectra is taken at normal incidence with the polarisation perpendicular to
the grating grooves.
their respective lattice points and of equal size. The portions of blue circle which lie
within the zero-order black circle represent diffracted orders which may couple out and
travel away from the surface, the diffracted light. The dashed lines represent the SPP
momentum states. The black dashed SPP cone cannot be coupled to by free-space light,
while the diffracted SPPs (blue dashed) which lie inside the zero-order light circle may
be coupled with free-space radiation. These
k
-space diagrams also indicate the direction
of the group velocity of the surface modes, which travel away from their respective
lattice points and always lie perpendicular to the SPP contour.
2.4.4 Coupling Strength of Light to SPPs on Gratings
A typical specular reflection spectrum for a metal diffraction grating is shown in figure
2.10 and demonstrates some typical features observed in the reflectivity of a grating
supporting SPPs. This spectrum, produced using the Chandezon theoretical modelling
method outlined in chapter 3, is for a sinusoidal grating with a pitch of
λ
gx
= 750 nm,
and depth of 40 nm with a frequency dependent dielectric function for silver determined
from literature [
84
]. The spectrum is calculated at normal incidence with the polarisation
perpendicular to the grating grooves.
25
2. Background Theory
Features of interest in this figure are labelled on the graph. The diffraction edge,
the wavelength at which the first diffracting order begins to (or ceases to) propagate
is clearly shown at 750 nm as a sharp critical edge. Slightly higher in wavelength, at
λ
0
765 nm, there is a sharp resonance in the reflectivity presenting as a minimum in
the reflected light. This minimum is a result of the reflected light being out-of-phase
with the light which is re-radiated from decaying SPPs. The incident light is scattered
by the grating and, when the coupling conditions are met, resonantly excites a SPP
which travels along the surface. This SPP may then decay and scatter back into reflected
light. A total of two scattering processes, an excitation and subsequent decay of a SPP,
give the total phase change of the re-radiated light of 90
+ 90
+ 90
+ 90
= 360
relative to the incident light, while the reflected light which did not interact with the
SPP has accumulated a total phase change of 180
by the simple reflection from a metal
surface. This results in the re-radiated light being out-of-phase with the reflected light
by 180
, causing the observation of a minimum in reflection. Since no light is reflected,
the energy is lost to Joule heating.
The position of the second order diffraction edge is marked on the spectrum, but
is very weak. Beyond this diffraction edge, at
λ
0
420 nm there is a small resonance
exhibited as a region of enhanced reflectivity, a ‘bright spot’. This is the result of
reflected light being in-phase with the light re-radiated from the coupling of a 2
nd
order SPP. The presentation of an optical interaction with a SPP as a maximum in the
reflectivity is again a result of the phase difference accumulated from the scattering,
coupling, and decay of light and the SPP. On a purely sinusoidal grating such as the
one modelled here, multiple scattering processes are required to couple to this SPP,
leading to the re-radiated light being in-phase with the non-interacting reflection. The
waves constructively interfere, resulting in a maximum in the reflectivity coefficient.
The drop in the reflectivity at lower wavelengths is due to the metal’s natural
absorptions due to inter-band transitions and bulk plasmon excitations. The imaginary
component of the metal’s dielectric function at these wavelengths reflect these absorptions
by becoming much larger. This ‘background’ absorption can be seen in all the silver
grating results in this thesis.
2.4.5 Plasmonic Band Gaps
When a propagating SPP encounters a counter-propagating SPP with equal energy
and equal and opposite momentum, they will constructively interfere producing SPP
standing waves [
35
,
85
]. Typically, two possible SPP standing waves can occur, with
the nodes and anti-nodes of one SPP standing wave shifted
λ/
4 in space with respect to
the other, a typical result of any quantised standing waves confined to a single value of
26
2. Background Theory
E
0
λ
g
2λ
g
3λ
g
(a)
E
0
λ
g
2λ
g
3λ
g
(b)
Figure 2.11: Cartoon of two possible SPP standing waves on a sinusoidal grating.
The blue and red regions show accumulation of positive and negative charge density,
respectively. These two standing waves, both which have a period of 2
λ
g
, result in
different field arrangements and so are associated with different energies.
momentum. This is analogous to the well known physical examples of electron standing
waves formed at Brillouin Zone (BZ) boundaries, or the vibrational modes of a solid
rod.
On a flat surface, these two surface standing waves of electron density and coupled
electromagnetic field are indistinguishable, save for the phase difference between them.
They occur at the same energy and momentum as one another. On a diffraction grating
the situation is quite different. The arrangements of charge along the shaped surface
can result in the electron surface charge density sitting in very different electromagnetic
potentials, for example at the bottom of a grating groove or at the top, which is
depicted in figure 2.11. In this case, the two possible standing waves have the same
period (enforced by the periodic lattice) and so the same momentum, but will be
associated with a different electromagnetic energy. Between these two energy values,
SPP propagation is forbidden, due to destructive interference of the surface waves. An
energy region of forbidden SPP propagation has been opened up, a plasmonic band-gap.
This is shown in the extended zone scheme dispersion plot in figure 2.12(a) which shows
the Bragg scattering of SPPs at the first BZ boundary (at
k
g
/
2) causing the formation
of a band gap.
The band gap shown in figure 2.12(a) will not be observed, as no available scattering
event can diffract this region into the zero-order light cone. However, the band gap
which occurs at the 2
nd
BZ boundary may be, as shown in the repeated zone scheme
dispersion in figure 2.12(b)
. In this case, while a single 1
k
g
scattering event may bring
In this figure, the previous band gap at k
x
= k
g
/2 has been omitted for clarity.
27
2. Background Theory
angular frequency, ω
in-plane wavevector, k
x
k
g
/2
0
k
g
/2
(a)
angular frequency, ω
in-plane wavevector, k
x
k
g
0
k
g
(b)
Figure 2.12: The dispersion and band gap of (a) a SPP on a grating with one Fourier
Harmonic and (b) on a grating with the first two Fourier Harmonics. The red line is
the light line for free space light, the blue curves show the SPP contours. In (a) the
black dotted line shown the unperturbed SPP contour, and the green dotted lines show
the position of the first BZ.
the band gap region into the radiative zone, the interaction which leads to band gap
formation is between a zero-order and the 2
nd
order SPP at the 2
nd
BZ boundary
(
k
x
=
k
g
). This means that the required scattering of the SPPs must be a 2
k
g
event for
band gap formation. On purely sinusoidal gratings, which possess no 2
k
g
component in
their grating profile, this scattering process is mediated by multiple scattering events,
which are very weak. Since this scattering amplitude for the SPP is so weak, the
interference between counter-propagating modes will also be weak, and band gaps are
unlikely to be large enough to be observed. If, however, the grating profile contains a
2
k
g
component (by, for example, the addition of a secondary pitch of the grating equal
to half the period of the fundamental period), the SPPs may couple together to form a
standing wave via a single strong scattering event. It is the 2
k
g
component that couples
the SPPs together to form the band-gap, and the 1
k
g
component that scatters this into
the radiative region and so may be observed, as shown in figure 2.12(b).
Again, this is analogous to the formation of the electron band-structure of crystals,
formed when considering nearly-free electrons in a periodic potential [
75
], or photonic
band-gaps in dielectric stacks/photonic crystals [86].
A subtle difference in the definition used in this thesis, and common in the research
on surface plasmons, is that a ‘plasmonic band-gap’ might only occur over a limited
28
2. Background Theory
range of angles, whereas for electron band structure or photonic crystals a ‘band-gap’
occurs in all possible propagation directions, and the term ‘stop-band’ is used for
single-direction gaps. Here, we adopt the convention of a band-gap occurring if an
energy gap exists for which SPP propagation is forbidden over a given angle range, and
reserve the term ‘full plasmonic band-gap’ for forbidden SPP propagation for a range of
frequencies in all directions.
2.4.5.1 Coupling Light to Band Edges
To scatter SPPs in to the radiative region and also couple the two SPP modes together
to form standing waves requires suitable grating harmonics in the grating profile. While
the magnitude and position of the band gap is determined solely by the presence and
size of these grating harmonics, the coupling of light to the SPP standing waves depends
strongly on the relative phase between the grating components. We can demonstrate
this simply by considering the band gap which will open at normal incidence (
θ
= 0
)
for a grating described by the simple Fourier sum,
f(x) = A
1
sin(k
g
x) + A
2
sin(2k
g
x + ψ
2
)
where, in our case,
A
1
= 5 nm,
A
2
= 2 nm,
k
g
= 2
π
g
where
λ
g
= 605 nm.
ε
=
17
.
5 + 0
.
7
i
.
ψ
2
is the relative phase of the 2
k
g
component relative to the
k
g
component.
Three reflectivity plots as a function of angular frequency and in-plane momentum
around the band gap region are shown on the right side of figure 2.13, with each plot
showing the numerically calculated reflectivity for different values of
ψ
2
. The method
used for calculation is the Chandezon method [
87
,
88
], detailed later in chapter 3. For
the first case,
ψ
= 0
, both band edges couple strongly to light, and are shown as a
minimum in the reflectivity, forming two symmetric blue bands. For
ψ
= +90
, only
the upper frequency (higher energy) is coupled, while for
ψ
=
90
, only the lower
frequency band edge is coupled.
The explanation for this phase dependent coupling is demonstrated in the figures
of the grating profiles and components, also shown in figure 2.13. The lower half
space of each of these sketches shows the relative phase between the two grating
components, and the upper half-space shows the resultant surface profile by addition
of the
k
g
and 2
k
g
components. Firstly, by symmetry, the nodes and anti-nodes of a
SPP standing wave on such a grating will occur at either the peaks or the troughs of
the 2
k
g
component. Alternating nodes and anti-nodes at these points provide a SPP
standing wave which satisfies the symmetry of the grating, and also has a wave vector
equal to (
k
g
(
k
g
))
/
2 =
k
g
at normal incidence (which is half way between the +
k
g
and
k
g
scattered SPPs). One SPP standing wave charge arrangement will require
29
2. Background Theory
gratingcomponents
x
k
g
2k
g
ψ
2
= 0
E
in-plane momentum, k
x
(m
1
)
angular frequency, ω (rad s
1
)
0.6
0.7
0.8
0.9
-1.5 -1 -0.5 0 0.5 1 1.5
2.98 3 3.02 3.04 3.06
× 10
5
× 10
15
(a) ψ
2
= 0
gratingcomponents
x
k
g
2k
g
ψ
2
= +90
E
in-plane momentum, k
x
(m
1
)
angular frequency, ω (rad s
1
)
0.4
0.5
0.6
0.7
0.8
0.9
-1.5 -1 -0.5 0 0.5 1 1.5
2.98 3 3.02 3.04 3.06
× 10
5
× 10
15
(b) ψ
2
= 90
gratingcomponents
x
k
g
2k
g
ψ
2
= 90
E
in-plane momentum, k
x
(m
1
)
angular frequency, ω (rad s
1
)
0.4
0.5
0.6
0.7
0.8
0.9
-1.5 -1 -0.5 0 0.5 1 1.5
2.98 3 3.02 3.04 3.06
× 10
5
× 10
15
(c) ψ
2
= 90
Figure 2.13: Sketches of different grating profiles determined by the relative phases
of the
k
g
and 2
k
g
components (left), and the resulting coupling of the light to the
band-edges, shown as modelled reflectivity plots (right).
30
2. Background Theory
induced charge at the peaks of the 2
k
g
component, while the other solution will require
the charge to accumulate at the troughs.
In order for light to couple to the SPP standing waves, the electric field vector
must have a component normal to the surface by which the impinging field may induce
surface charge and hence couple to the surface electron density oscillations. In figure
2.13(a), the addition of the in-phase (
ψ
= 0
) 2
k
g
component leads to points for both
the high energy (green dots) and low energy (blue dots) nodes and anti-nodes which lie
on the grating profile where the gradient of the profile is not zero. Because the grating
profile at these points is not flat, there is indeed a surface normal component which
can match to the impinging electric field, and the result is that both the high energy
and low energy solutions may couple to incident light. In the second case in figure
2.13(b), where
ψ
2
= +90
, the resulting surface profile only has a non-zero gradient for
the high energy solution (green dots), while the position of the low-energy ‘hot-spots’
(the 2
k
g
minima) occur at a flat region of the grating. Consequently, no coupling
occurs for the lower-energy mode, and only the upper band is coupled, as seen in the
reflectivity plot. Finally, for the
ψ
2
=
90
case in figure 2.13(c), the resulting geometry
provides a normal component of the surface to the electric field only at the blue dot
locations, corresponding to an induced charge arrangement for the low-energy band.
The high-energy arrangement points are located at the 2
k
g
component’s maxima, and
exist on flat regions of the grating geometry. The coupling of light in this case is then
only to the lower-energy mode.
2.4.6 Polarisation Conversion
Linearly polarised light incident on a grating may be converted to the orthogonal
linear polarisation state in a process named ‘polarisation conversion’. Two mechanisms
have been identified for this conversion using diffraction gratings; one is mediated by
SPPs, and the other is due to the phase lag of the different reflected polarisation state
reflections on deep gratings.
SPPs may mediate the polarisation conversion of incident light when the excited
SPPs travel along the grating surface in a direction of broken symmetry [
26
,
34
,
89
,
90
].
For the monograting case, this is any azimuthal angle other than
φ
= 0
or 90
. In such
an orientation, both TM and TE polarised light provide a normal component of electric
field to the grating surface and so may induce surface charge and resonantly drive
SPPs. Since this is the case, it is clear that SPPs decaying back into propagating light
may couple back out into either the TM or TE polarisation state, and so polarisation
conversion is possible. For shallow gratings, the maximum polarisation conversion occurs
at an azimuthal angle of
φ
= 45
, where the coupling to SPPs with TM and TE polarised
31
2. Background Theory
light is equal. For shallow gratings, the polarisation converted signal is proportional to
sin
2
(2
φ
), which is shown simply by the consideration of the the electric field components
relative to the grating surface [
34
]. For bigratings, the same considerations hold true. A
SPP can mediate polarisation conversion when travelling along an axis with no mirror
symmetry, and cannot otherwise. For example, at
φ
= 45
on a square bigrating, no
polarisation conversion is expected.
The second method by which diffraction gratings may exhibit polarisation conversion
is again due to broken symmetry between the two orthogonal polarisation states [
59
].
This process is not mediated by SPPs and may occur on sub-wavelength, non-diffracting
gratings providing the grooves are sufficiently deep. Consider the simple case of a
non-diffracting monograting, the electric field incident on the grating surface may be
considered to decompose in two directions, one parallel to the grating grooves (
E
k
) and
one perpendicular (
E
) to the grooves. Since
E
k
does not cut across the grating grooves,
this field will reflect as if it were reflected from a planar surface at a position equal to
the average plane of the grating. However, since
E
cuts across the grating grooves,
circulating fields may be produced in these grooves which alter the effective average
plane of reflection for this field component. The difference in position of these ‘effective
mirrors’ produces a phase difference between the two components which serves to rotate
the plane of polarisation, leading to polarisation conversion. The maximum polarisation
conversion on such gratings occurs when the phase difference between the reflected
E
k
and
E
is 180
, and so is a cyclic function with respect to the depth of the grating.
This necessity for the field components to be 180
out of phase requires the gratings to
be reasonably deep. For a gold sinusoidal grating illuminated with
λ
0
= 632
.
8 nm with
a pitch of 300 nm the maximum polarisation converted signal occurs at a groove depth
of 180 nm [59].
32
Chapter 3
Theoretical Methods
3.1 Introduction
The calculation of the optical response of diffraction gratings has been an active pursuit of
electromagnetic research for over a century. In 1907, Lord Rayleigh proposed a treatment
for the reflected light from periodic surfaces by considering them a perturbation of the
case of a flat surface [
7
]. This theory calculated that there must be a sharp discontinuity
in the reflectivity from the grating when a diffracted order of light emerges from, or
disappears beneath, the grating surface. This discontinuity, first observed experimentally
as a dark band by Wood [
5
], corresponds to the instances at which power must be
redistributed to the propagating orders of light as a diffracted order either starts, or
ceases, to propagate at the grating surface. However, Rayleigh’s theory was inadequate
for all but the shallowest gratings, as it lacked a full consideration for the fields that
could occur in the grating grooves. Furthermore, Rayleigh’s treatment left the bright
band observed by Wood, characteristic of a SPP excitation, unexplained.
Since 1907, many methods by which to model the optical response of gratings have
been developed and have been verified experimentally. Methods such as Finite Difference
Time domain methods and Fourier modal-matching methods have all been tested. A
review of the these, and other, numerical methods of grating analysis can be found in
reference [36].
The purpose of this chapter is to provide a background to the methods used in
this thesis, namely the Finite Element Method (FEM) method and the coordinate
transformation method of Chandezon et al. [87].
33
3. Theoretical Methods
3.2 The Differential Method of Chandezon et al.
The method of Chandezon et al. [
31
,
87
] solves Maxwell’s equations by utilising a
coordinate transformation which maps the grating interfaces on to flat, parallel, planes.
This avoids the Rayleigh theory’s failure to account fully for field inside the grooves, as
the grating is effectively flattened by the coordinate transformation and the boundary
conditions are then matched across this flat interface. For a monograting the transformed
coordinates are,
u = x (3.1)
w = y (3.2)
v = z S(x) (3.3)
Where
S
(
x
) is the interface profile of the grating. The ease of matching the boundary
conditions in this modified frame is achieved at a cost of the need for more complex
expressions for Maxwell’s equations in a non-orthogonal coordinate system of (
u, v, w
).
In this method, the derivation of the incident and scattered fields from Maxwell’s
equations are expressed as an infinite series of plane waves, a Fourier expansion.
In order to solve this problem numerically, the infinite Fourier series must be
truncated, placing a limit on the number of scattered field components in the model. The
order of truncation is chosen from a compromise between obtaining suitable convergence
of the solution and minimising computation time. In this thesis, we primarily use
Chandezon method modelling to provide illustrative examples of reflectivity by which
we may explain certain observed optical features. For this purpose, we find that limiting
the number of scattered fields in the system to 6 is sufficient.
The one-dimensional case was presented by Chandezon in 1980 [
87
]. Since then, it
has been extended to multi-layered gratings [
91
,
92
] and to bigratings [
93
]. The method
used in the thesis is the multi-layer bigrating method reported by Harris et al.[
93
]. This
allows for two crossed gratings of different pitches to be considered, and the coordinate
transformation is altered from the equations 3.1, 3.2 and 3.3 to,
u = x (3.4)
w = y
0
= x cos α + y sin α (3.5)
v = z S(x, y
0
) (3.6)
Where
α
is the angle between the two crossed gratings. This interface profile
S
(
x, y
0
)
is represented by a Fourier sum. Because the technique is a differential method, sharp
discontinuities in the surface profile gradient are difficult to model. The number of
34
3. Theoretical Methods
Fourier components required to correctly simulate a step-like grating profile can become
prohibitively large, although extensions to the method have shown than the method can
be applied well to trapezium shaped grooves [
94
]. For a precise description of a lamellar
grating as produced by our fabrication techniques a large number of Fourier harmonics
are required to represent
S
(
x, y
0
), which can make computation times prohibitively long.
However for the grating periodicities used in this thesis only the dominant scattering
associated with low order Fourier components need be considered to gain sufficient
insight into the underlying physical processes.
The Chandezon method has been used successfully to probe the electromagnetic
response of hexagonal bigratings [
44
], the effect of groove shape on reflectivity resonances
due to SPP excitations [
33
,
42
], to examine asymmetric excitation of SPPs on blazed
gratings [
95
] and to determine the surface profile of gratings from experimentally
acquired reflectivity measurements [43].
3.3 The Finite Element Method
The finite element method (FEM) for predicting the electromagnetic response of systems
is a numerical modelling process that subdivides a system into discrete elements, solving
Maxwell’s equations for each element and ensuring the solutions are consistent across
each elementary boundary [
36
,
96
]. Broadly, the process iteratively applies three steps
until a solution is found:
1.
Determining the distribution and size of elements across the model. The continuous
array of elements is referred to as the ‘mesh’ of the model.
2.
Numerically solving Maxwell’s equations for each element and for the mesh as a
whole.
3.
The evaluation of the quality of the solution, to see if the model converges to a
consistent and physically accurate solution. If not, the process repeats from (1),
further refining the mesh.
The software package used for FEM modelling in this thesis is the HFSS package
from Ansys, Inc. [
97
]. The software includes a computer aided design component,
which allows the building of three-dimensional models to represent the sample under
consideration. These 3D models may be assigned different optical parameters, such as
frequency-dependent permittivities and permeabilities, and may be parametrised for
automated parametric alterations. The software’s default element shape is a tetrahedron,
with many tetrahedra forming a conformal mesh in the problem domain. In the case
of 3D models which contain curved surfaces, the software may also use curvilinear
35
3. Theoretical Methods
tetrahedral elements, greatly reducing the number of elements required to approximate
a curved surface.
3.3.1 Solving Maxwell’s Equations in the Mesh
The FEM package attempts to solve for the electric field [
98
],
E
, for each element using
the wave equation,
×
1
µ
× E
k
2
0
εE = 0 (3.7)
where
k
0
is the free space wavenumber and
ε
and
µ
are the relative permittivity and
relative permeability, respectively. As a numerical simulation, the software attempts
to minimise the left hand side of equation 3.7 to be as close to zero as possible for
each element, it will not in general, equal exactly zero. The magnetic field,
H
is then
calculated using,
H =
1
ωµ
× E (3.8)
where
ω
is the angular frequency of the wave. The values for
E
and
H
must be found
for all the elements forming the conformal mesh in the problem space, and must be
consistent across each tetrahedron boundary. This leads to thousands of simultaneous
equations in the form of equation 3.7 for each finite element in the model, which are
combined into a matrix and solved for E numerically.
Once the fields have been determined, a generalised scattering matrix (S-Matrix) is
derived for the system, from which initial and final optical properties such as reflectivity
or transmission can be extracted. The software may also attempt to find the frequencies
at which the poles of the S-Matrix occur, and so calculate the eigenmodes of the system
and their associated fields.
3.3.2 Adaptive Meshing
The structure of the mesh used in the FEM model is critical to obtaining an accurate
simulation. In general, a higher resolution of mesh is required in areas where the electric
(and magnetic) field vary rapidly. To achieve a suitable mesh, the FEM package uses a
process of adaptive meshing, refining the mesh in areas where the error in the electric
field calculation are high.
In the process, the software generates an initial, geometrically conformal mesh, and
solves equation 3.7 for the electric field for each of the tetrahedra. Based on this initial
solution, the area of the problem domain where the exact solution has a large amount
of error is determined [
98
]. This is done by examining the residuals of equation 3.7,
36
3. Theoretical Methods
such that the approximate solution, E
approx
gives,
×
1
µ
× E
approx
k
2
0
εE
approx
= residual (3.9)
for each tetrahedron. A set percentage of tetrahedra with the largest residuals are then
refined by sub-dividing the element into even smaller tetrahedra. Another solution
is then found with this new mesh, and the process is repeated until the convergence
criteria has been met, or the total number of adaptive passes, set by the user to prevent
overly complex and memory intensive meshes, has been met.
It is important to note that the simple addition of elements to the problem is
inefficient, and that the software’s determination of the refinement areas by consideration
of highest tetrahedral residuals is not foolproof. Occasionally, the software may attempt
to refine the mesh in locations that are not critical, but do lie in the upper percentile of
tetrahedral locations with high residuals. Users can make the simulation more efficient
by defining the initial mesh conditions using mesh operations in the software. For
example, someone setting up a simulation for plasmonic resonances in thin cavities may
start the initial mesh with a greater density of elements in the grooves (where they
expect rapidly varying fields) than the software alone might allocate.
3.3.3 Solution Evaluation and Convergence
The accuracy of the solution is determined by the model’s ‘convergence’. Because of
the direct relationship between the electric fields for a solution and the corresponding
S-Matrix, the convergence is measured by the change in the magnitude of two successive
solutions and is called the
S
value. It is a measure of the change in electric field
distribution between solutions. When the
S
values becomes less than a user-specified
amount, the model is deemed to be ‘converged’ and the final solution is obtained. An
example of mesh refinement is shown in figure 3.1. Here, we find that the refinement
algorithm adds more elements to the mesh, to a total of over 40,000 in the final case,
in order to reduce the
S
value (red line) and achieve convergence. This particular
example is the convergence for a zigzag grating model, the results of which are detailed
in chapter 7. For the results in this thesis, the maximum S value used was 2%.
3.3.4 Model Boundary Conditions
In the FEM package, the modelled system must be surrounded by appropriate boundary
conditions so a solution can be determined. Diffraction gratings in this thesis are
approximated to infinite periodic structures, as the propagation lengths of SPPs is
far smaller than the sample size. Thus, a unit cell of the structure with appropriate
37
3. Theoretical Methods
2 4 6 8 10 12
number of adaptive passes
number of model tetrahedra
0 1 2 3 4
× 10
4
0 1.99 3.98 5.97 7.96
ΔS (%)
Figure 3.1: A graph showing the convergence of a typical FEM model as a function of
adaptive passes. The black line shows the total number of elements introduced in the
model, while the red line shows the S parameter as a percentage.
boundary conditions is suitable for a full solution to the problem, an example being
shown in figure 3.2. This unit cell is surrounded by 4 types of boundary, labelled in the
figure. The first type is called a master-slave boundary. This boundary pair constrains
the electric and magnetic field so that fields at the master boundary must match exactly
the field at the slave boundary. The periodic boundary conditions imposed by the
master-slave pair ensure that the solution is exactly equivalent to an infinite array of unit
cells. The slave boundary may also replicate the fields at the master with an additional
relative phase delay. This phase delay is how the software calculates the fields for an
angle of incidence other than normal with the phase delay calculated internally from
the provided polar and azimuthal angles. The next type of boundary is the ‘Perfect
E’ boundary. This boundary acts as a perfect lossless conductor. It is placed on the
bottom of the metal film, and acts as an easy way to double the thickness of the metal
film by reflecting perfectly any impinging radiation. Since the field penetration for
visible radiation in to a metal such a silver is very small (on the order of 20 nm), a
metal film thickness of 50 nm, used together with the Perfect E boundary, provides a
suitable approximation to a semi-infinite metal substrate without the additional cost of
increasing the number of tetrahedra. The final boundary is a Floquet port. This acts
38
3. Theoretical Methods
Master/Slave 1 Master/Slave 2
Floquet port
Perfect E
Figure 3.2: Schematic of the different boundaries used in the FEM model. The unit
cell shown is a zigzag grating (chapter 7) in red. The boundaries are highlighted and
named in each case. The Perfect E boundary has been extruded from the bottom of the
model for clarity.
as a radiation boundary that is open to free-space. The FEM package actually absorbs
any radiation that hits this boundary, without reflection, mimicking the character of
radiation escaping to free-space. This Floquet port also acts as an excitation source
and a way by which certain orders of diffracted field are extracted, which is covered in
the next subsection.
3.3.5 Floquet Type Excitation
When extracting the reflectivity of a diffracting sample from the S-Matrix, it becomes
clear that simply examining the total field arriving above the sample does not correspond
meaningfully to the types of experiments typically conducted in this thesis. This is
because the modelled fields would contain the zero-order reflection and all the diffracted
orders simultaneously, while experimentally we would only usually collect light from
the zero-order reflected beam. Floquet analysis allows us to extract the appropriate
order of diffracted light by recognising that the reflected light will be a superposition of
all these possible fields, whose in-plane momentum will be totally defined by the polar
angle of incidence and the diffracting periodicities. The wavevectors of waves arriving
39
3. Theoretical Methods
above and at a plane parallel to the sample surface will be of the form,
k
k
= k
0
sin θ + nk
gx
+ mk
gv
(3.10)
Where
k
k
is the parallel wavevector,
k
0
is the free-space wavevector.
k
gx
and
k
gv
are the grating wavevectors in the
x
and
v
directions, defined as
k
gx
= 2
π
ˆ
x
x
and
k
gv
= 2
π
ˆ
v
x
. Depending on the diffracted order, they have the integer multiples
n
and
m
, so, for example, the first-order diffracted field in the
x
direction would be
n
= 1
and m = 0.
Diffracted waves reflected from, or transmitted through, a sample must have well
defined periods parallel to the surface and are determined by the diffracted orders, such
that,
E
n,m
(k
0
) = E
n,m
(k
0
+ nk
gx
+ mk
gv
) (3.11)
Where E
n,m
is the (n, m)
th
electric field diffracted order.
The FEM package calculates all the possible propagating orders, based simply on
the periodicity of the surface and the wavelength of the incident field, and is then able to
extract from the S-Matrix each field solution corresponding to each order. Because the
periods of the fields in the simulation are now restricted, if there are strong effects from
evanescent, non-propagating orders, (such as trapped SPPs) these must be included
as well. The calculation of which ‘Floquet modes’ to include is undertaken at a single
frequency, so it is important to calculate these using the highest frequency that is to be
simulated, to include all possible diffracted orders. The choice of the number of Floquet
modes to include is the equivalent of determining how many Fourier components of light
or of a surface to include for an accurate result, much like in the Chandezon case.
40
Chapter 4
Experimental Methodology
This chapter details the experimental methods used in this work for the fabrication and
measurement of metallic diffraction gratings supporting SPPs. Section 4.1 describes
the fabrication method for the gratings, a combination of electron beam lithography
(EBL) and template stripping. The second section explains the established methods by
which the reflectivity of gratings was measured as functions of both wavelength and
angle. Finally, section 4.3 introduces a new experimental method for the mapping of
SPP iso-frequency contours in momentum space, which was developed as part of this
work and has not been reported previously in the literature.
4.1 Sample Fabrication
The production of diffraction gratings used in this thesis has been achieved using a
combination of electron beam lithography and template stripping. Briefly, this process
involves the exposure of resist to a diffraction grating pattern using a focussed electron
beam. This resist, when developed, is used as an etching mask for the silicon substrate,
producing a master template in a silicon ‘chip’. This chip is then metallised and the
pattern is transferred to a glass substrate by gluing the metal surface to the glass
substrate and removing the master with a razor blade. This leaves an inverse copy
of the master grating on the glass substrate, and the silicon master may be reused to
produce further copies.
An outline of the process is shown in figure 4.1. The details of the process are
explained in depth in the following sections.
4.1.1 Electron Beam Lithography
All electron beam lithography steps are performed in an ISO class 6 cleanroom. This
minimises the risk of small particle contamination and unwanted organic residues being
41
4. Experimental Methodology
electron beam
PMMA
Silicon Master
Reactive Ion Etch
3. Produced Master in Silicon
4. Thermal Evaporation of Silver
Glue
Glass Superstrate
Template Strip
Fabricated Grating
Glass Substrate
1. Electron Beam Lithography
2. Reactive Ion Etch
5. Superstrate Adhesion
6. Template Strip 7. Final Sample
Produced
Figure 4.1: Illustration of the fabrication method used for the production of diffraction
gratings.
42
4. Experimental Methodology
introduced to the samples.
A silicon wafer is prepared as a master substrate. Any large-particle dust is removed
from the wafer surface using pressurised nitrogen gas. The wafer surface is then spin-
coated in a
600 nm thick protective layer of Poly(methyl methacrylate) (PMMA). The
typical protective polymer used was PMMA 950K A6, spun at 2000 RPM for 40 seconds
to produce the 600 nm layer. The wafer was then diced into 1 mm
2
chips which would
be used as the grating master substrates. The resist layer protects the chip surface from
the silicon dust that the dicing process produces. This layer, and the accompanying
contamination, is removed by inverting the chips in 100 ml of warm acetone for 10
minutes. After removing the protective resist, the chips are placed into a fresh solution
of boiling acetone (80
C) for 1 hour, and then sonicated in 100 ml of isopropanol for 1
hour. The substrates may occasionally splinter during sonication, (introducing unwanted
Si dust to the solution), these chips are discarded. After 1 hour of sonication the chips
are removed from the isopropanol and dried using pressurised nitrogen gas. The clean
wafer chips are then inspected under dark-field optical microscopy to ensure the surface
is optically clean. If not, the cleaning process is repeated.
The electron resist is then added to the surface. The resist used in all grating
fabrications included here is PMMA 950K A4. This is a positive electron-resist, whereby
exposure of the resist to the electron beam causes de-cross-linking of the polymer chain,
allowing the removal of the exposed resist with developer. The chip is placed in a
motorised spincoater, and 3-4 drops of PMMA are introduced to the surface using a
new, clean glass pipette. The chip plus resist is then spun at 4500 RPM for 40 seconds
to leave a 180 nm thick layer of resist on the surface [
99
]. The substrate is then baked
at 165
C for 10 minutes on a hot-plate. This baking raises the resist temperature above
the glass transition temperature, and also causes the evaporation of any remaining
solvent (Anisol), improving substrate adhesion.
The substrate is then loaded into the electron beam lithography system and the
grating pattern is exposed. The system used is a nB4 Electron Beam Lithography
system (NanoBeam Limited [
100
]) and exposures are performed with a beam current of
2 nA
and a beam accelerating voltage of 80 keV. The exposure dose ranged between
400 600 µC cm
2
, depending on the grating. Initially, the grating pattern was exposed
four times at different locations on one substrate, each grating using a different test dose.
The final samples produced from these test areas are examined using scanning electron
microscopy and the grating’s electromagnetic response is measured using the optical
techniques listed in section 4.2, in order to determine the most suitable electron dose
to use for each particular grating. Once a suitable dose has been determined, future
gratings are produced with one grating per chip.
The electron beam write field size, (the region exposed before the substrate needs to
43
4. Experimental Methodology
be moved) was
100 µm
, to minimise the effect of height variation across the substrate.
This is the maximum write field available in the system used. The nB4 electron beam
system has a theoretical beam diameter of
2.3 nm
at 100 keV, and the stitching error
between write-fields is on the order of 20 nm. This stitching error introduces an
extremely weak long-pitch periodicity to the samples, on the order of the write field
size. The weak diffractive properties from these stitching errors are not found to inhibit
the optical efficiencies of the gratings significantly, and are not observed experimentally.
The exposed sample then requires development to remove the de-cross-linked poly-
mer and provide a polymer etching mask for the Si. The developer used is a 15:5:1
solution of IPA:Methyl isobutyl ketone:Methyl ethyl ketone (IPA:MIBK:MEK) [
101
].
An endothermic reaction occurs when this solution is first mixed, so to ensure devel-
opment times remain consistent between processes the developer must be allowed to
return to room temperature before use. The exposed chip is submerged in the solution
for 1 minute, followed immediately by a further 1 minute in a beaker of IPA. When
transferring the sample between the developer and the IPA, it is advisable to maintain a
small amount of the developer in a droplet on the sample surface, to prevent premature
exposure to the air. After the 1 minute in IPA, the sample is removed and blow-dried
with nitrogen gas. At this point the diffraction properties of the sample will be apparent,
and illumination with an appropriate wavelength laser allows an estimation/check of
the grating’s periodicity and orientation.
At this stage, the grating pattern has been transferred to the resist and successfully
developed. The sample consists of a coating of un-exposed PMMA layer, in which a
180 nm deep pattern has been developed. This polymer grating sits on a Si substrate.
The pattern is now transferred to the Si chip by means of reactive ion etching (RIE),
using the PMMA as a polymer mask. The reactive ion etching gas used is an oxygen,
CHF
3
mix, with the following parameters: 100 sccm
CHF
3
, 3 sccm O
2
, at
38 mTorr
of pressure and 100 W power. This gives an etch rate into the Si of approximately
29 nm min
1
The selectivity of Si to PMMA etching is sufficient for a 180 nm thick
layer of PMMA to be a suitable etching mask for depths of up to 80 nm etched into the
Si substrate. After RIE, any excess resist is removed by boiling the sample in acetone
and sonication in IPA. This leaves a master grating pattern in a Si substrate.
For ‘crossed’ bigratings, where a second periodicity runs along the grating surface at
an angle to the first periodicity, this process is repeated with a previously fabricated Si
master grating used in place of the flat Si substrate. This double exposure method was
preferred as the result is a grating where the two amplitudes have been summed, which
is consistent with the theoretical modelling undertaken with the method of Chandezon
standard cubic centimetre per minute
44
4. Experimental Methodology
introduced in chapter 3.
4.1.2 Thermal Evaporation
To produce metallic diffraction gratings from the Si master, the master substrate is first
metallised using thermally evaporated silver under high vacuum.
The clean template is placed in a vacuum chamber facing a molybdenum boat
containing 99.999% pure silver. The pressure in the vacuum chamber is reduced to
5
×
10
6
mbar by use of both a rotary pump (down to 1
×
10
2
mbar) and a water-
cooled turbo pump. At this low pressure, current is passed through the boat, which
heats, via Joule heating, to a temperature sufficient for the silver to evaporate, with
the silver vapour then condensing on to the structure. The rate at which the silver
evaporates is controlled by the current passing through the molybdenum boat. A typical
driving voltage of 70 V produces a deposition rate of approximately 3
˚
A
s
1
. The rate
of deposition and the nominal thickness is measured using a quartz crystal thickness
monitor.
The depth of the features for a grating produced by template stripping (described
in the next section) is pre-determined by the silicon master’s depth that was etched
via RIE. The amount of silver deposited on the master must then be greater than the
depth of the structures to ensure the sample is optically thick. It is found that for the
nominal depths of grating produced (ranging from 30 nm to 80 nm), a 300 nm thick
layer of silver was sufficient to ensure good pattern transfer and optically thick samples,
without producing unwanted silver film tension that may cause the silver to flake as the
layer relaxes during cooling.
4.1.3 Template Stripping
The method for transferring the master pattern to a glass substrate is based on the
method of Nagpal et al [
102
,
103
]. A glass substrate is prepared by first swabbing the
surface with an acetone soaked cotton bud to remove any organic residue. The surface is
then drag-cleaned using lens tissue and IPA, removing any larger dust/cotton particles
before being examined under an optical microscope.
The metallised Si template is then positioned on a flat, clean surface and a small
amount of UV optical adhesive (Norland 60 or 65) is introduced to the metallised
patterned area using a micro-pipette tip. Great care must be taken to ensure no
air-bubbles are present in this adhesive polymer, so it is advisable to de-gas the glue
first by agitation in a UV free environment. The clean glass substrate is then placed
gently on the master and the glue is allowed to spread across the interface
. Without
If too much adhesive has been added, the glue may reach areas of the Si chip which are not
45
4. Experimental Methodology
moving the sample, the adhesive is cured under incoherent UV light provided by a
mercury lamp. The polymer is cured for a total of 20 minutes under this light
.
After curing, the sample is turned over and a clean, sharp razor blade is inserted
between the glass (now the substrate) and Si chip (now the superstrate). With a small
amount of force, the master Si chip will be pried free and removed from the glass,
leaving behind the silver metal and, in it, the inverse metallic pattern. Once the Si
chip is freed from the glass/silver surface, the razor blade should be supplanted with
soft-tipped tweezers and the chip removed carefully, so as not to scratch or damage the
fragile silver surface.
Since silver degrades in the laboratory environment due to sulphur present in air,
this last step can be delayed until the sample is to be used. Until this point, the grating
pattern remains embedded in the Si/Silver interface, protected from the air and any
other possible contaminants.
Comparison between the master and sample SEMs shows less than 2% difference
between their mark-to-space ratios, indicating the high precision of copies taken from
the silicon master. The small difference may be attributed to the thermal relaxation of
the silver.
4.1.4 Polymer Replication
The embedding of gratings in a glass-like material can be desirable if the experimenter
wishes to access modes that may be lowered in energy in to the visible regime by use of
a higher refractive index. This is the case for a sample in chapter 7, and so the details
of embedding the grating in a high index material is covered briefly here. The Si master
of the grating, prepared via EBL, and RIE, is coated with a UV curable polymer and a
thin, flexible polymer sheet (acetate sheet). This combined polymer sheet and glue is
placed under pressure and cured rapidly using a high intensity UV source. The polymer
replica is then peeled off the master, and coated with a second UV curable polymer
(Norland 63). A clean glass slide is placed on top of this arrangement, and the glue is
cured, again by a UV light source. The polymer-master is peeled off the glass substrate,
leaving a polymer-replica on the glass substrate. This is then metallised via thermal
evaporation with a thickness of 300 nm of silver, and the bare side of the glass substrate
is adhered to a hemi-spherical prism using index matching fluid. The resulting sample
is illuminated through the prism/index matching fluid/glass/grating side, allowing for
metallised. In this case the template strip will be unsuccessful, and the process must be abandoned
at this point. The glass substrate is removed with great care and the master cleaned carefully by
submerging it in Acetone and then IPA.
It is useful to include the pipette under the UV light also, as the residual glue present provides a
secondary check that the polymer is fully cured before attempting the template strip.
46
4. Experimental Methodology
the investigation of a grating sample embedded in a high index of n = 1.52.
4.2 Angle Scans
4.2.1 Monochromator
M1
MC
M2
M3
M4
P
C
Pol
L
BS
D1
Sample
D2
Pol
Figure 4.2: System for the measurement of the reflectivity of a sample as a function
of both wavelength,
λ
0
and polar angle
θ
. White light is represented as blue lines,
pseudo-monochromatic light is shown as green lines.
To record the optical response of the diffraction gratings as a function of both incident
wavelength,
λ
0
, and polar angle,
θ
, a conventional rotational stage/monochromator
system is used. A schematic of the equipment is shown in figure 4.2. The incident
wavelength is selected by directing a white light source through the monochromator via
the focussing mirror, M1. Using an internal metallic diffraction grating and an output slit,
the monochromator outputs a beam of almost monchromatic light with a spectral width
of
λ
0
±
0
.
25 nm, with a selectable wavelength range between
400 nm < λ
0
< 850 nm
.
This light is then collimated by a pair of mirrors, labelled M2 and M3 in figure 4.2, and
directed down the primary optical axis by a plane mirror located at M4. The light is
then spatially filtered by a 0.5 mm pinhole (P) and passed through an optical chopper
(C) operating at approximately 1
.
3 k
Hz
. The chopper’s operating frequency is passed to
a pair of phase-sensitive detectors and is used as the triggering frequency for the signals
obtained from the detectors (D1 and D2) that are photomultiplier tubes. The beam is
then passed through a linear polariser (Pol) with which the polarisation dependence of
the optical response may be investigated.
For small-area samples, such as diffraction gratings fabricated using EBL, the grating
area may be as small as 2 mm
2
. In order to ensure the entire optical field impinges
the grating, and not the surrounding material, the beam is focussed with a lens, L,
with a focal length of 0.5 m. The distance between the pinhole, P, and the lens, L,
is approximately 1 m, approximating to a collimated far-field image with a radius of
47
4. Experimental Methodology
HeNe
C ND
Pol
LBS
Sample
D2
D1
Figure 4.3: System for the measurement of angular dependant reflectivity at a set
wavelength of λ
0
= 632.8 nm
0.5 mm, which is reproduced at the lens’s focal point. For smaller beam-spot areas, a
smaller pinhole may be used, at the cost of light intensity.
A small fraction of the light is directed to a reference detector, D1 via a plane-glass
beam splitter, BS. This reference signal allows for the normalization of the time-
dependence of the white-light source intensity. The transmitted beam continues along
the primary optical axis and hits the sample positioned precisely on a robotic rotational
stage, such that the sample position is central to the rotational axis and also at the
focal position of lens L. The robotic table, controlled via computer, sets the polar angle,
θ
of the sample, with the spectral reflectivity detector, D2, mounted so that it moves a
corresponding angle of 2θ.
The spectrally reflected light is collected by the detector D2 after passing though a
second linear polariser. Both detectors, D1 and D2, are photomultipler tubes whose
output current varies linearly with incident light intensity. The signals from the detectors
are passed to a set of phase sensitive detectors which, together with the chopper triggering
signal, extract the reference and reflectivity measurement and pass the information to
the computerised recording software via an analogue interface card.
To obtain a reflectivity spectrum of a sample, the output voltages from both the
reference and reflection PSDs are recorded as a function of both angle and wavelength.
To normalise the resulting dataset, the sample is removed and the light directed into
the reflectivity detector at
θ
= 0
. The spectra of this ‘straight-through’ reference signal
is recorded, and then the original dataset is divided by the reference spectra, obtaining
the normalised reflection spectrum of the sample as a function of both wavelength and
polar angle.
4.2.2 Fixed Wavelength Scans
A similar set-up is used for acquiring angle dependent reflectivity of samples at a single
wavelength. In this set-up, shown in figure 4.3, a HeNe laser (
λ
0
= 632
.
8 nm) is passed
through an optical chopper, C, a neutral density filter, ND, and a polariser before a
48
4. Experimental Methodology
Figure 4.4: Exploded view of the sample mount for use with a glass hemisphere. The
hemisphere is adhered to the glass substrate using index matching fluid. The rig allows
azimuthal rotation of the sample and hemisphere.
beamsplitter (BS) directs a small fraction of the beam to a reference detector. The
beam transmitted through the beam splitter continues along the optical axis and is
passed through an optional lens before impinging on the sample placed, again, at the
center of rotation of a robotic table. The detectors used are photo-diodes, whose voltage
output is linearly dependent on the incident intensity. The signal from the detectors are
passed to a set of phase sensitive detectors, and the signal extracted, as before, using
the mechanical chopper as a triggering reference.
4.2.3 Embedded Samples
Some samples presented in this thesis are investigated with the grating embedded in a
high-index substrate. For such samples, a glass hemisphere arrangement is used (Figure
4.4). This allows illumination of the sample without the need for any corrections to the
polar angle which might otherwise be necessary because of refraction. The sample is
affixed to the hemisphere using index-matching fluid.
4.3 Iso-Frequency Contour Measurement Using Scatterom-
etry
A desirable measurement of the dispersion of SPPs on diffraction gratings is to record
the mode’s iso-frequency contours in reciprocal space. By mapping the position of
49
4. Experimental Methodology
0 k
x
k
y
ω
k
gx
k
gx
k
x
k
gx
k
gx
ω
2
ω
3
ω
1
Figure 4.5: Three iso-frequency contours for the intersection of two SPP cones at angular
frequencies
ω
1
, ω
2
and
ω
3
. The contours show how the band-gap presents in
k
-space for
the cases of a frequency lower than the band-gap (
ω
1
), inside the band-gap (
ω
2
) and
above the band-gap (ω
3
).
the SPP contours in
k
-space, information is obtained about the 2D plasmonic crystal
lattice and the interaction of scattered SPP modes on the surface. These iso-frequency
contours are obtained as ‘slices’ through the 3D dispersion plot (
ω, k
) of the SPPs on a
grating for a constant frequency, and so are often referred to as equi-energy contours.
Examples of such iso-frequency contours are shown in figure 4.5. The upper figure shows
the intersection of two SPP dispersion cones (black lines), one scattered by +
k
gx
and
the other by
k
gx
, plotted in the
ω k
x
plane. At
k
x
= 0 these two SPP dispersion
curves meet and form a band gap, as shown in the figure. The dotted black line shows
how the SPP dispersion would have continued with no interaction. Three slices of
iso-frequency are shown,
ω
1
, ω
2
and
ω
3
, each chosen to illustrate a different SPP contour
50
4. Experimental Methodology
type. The corresponding SPP contours in kspace are shown in the lower figure. The
frequency
ω
1
is below the plasmonic band-gap in the dispersion, and the SPP cones
trace two distinct, approximately circular, contours in
k
-space. For
ω
2
, the energy
slice is taken in the band-gap and the resulting SPP contours have ‘merged’, with
no SPP solution along the
k
y
= 0 line around the intersection. Finally,
ω
3
shows an
intersection of the dispersion cones above the band-gap. This results in two contours,
one a larger merged ‘peanut’-shaped contour and a smaller SPP contour based around
k
x
= k
y
= 0. These SPP contours are analogous to iso-frequency Fermi-surfaces found
in solid state physics. This simplified figure is not periodic as it only includes two
scattered SPP cones for clarity, but the features observed for each energy slice will be
similar in all cases of SPP contour measurement. These contours show the presence of
band-gaps, the direction of SPP propagation (since the group velocity lies normal to the
contour) and gives information about the underlying scattering lattice. The complete
three-dimensional band structure of SPPs on periodic systems may be mapped by
collating the iso-frequency contours for a range of frequencies. The ability to map these
iso-frequency slices in momentum space is hence a powerful tool in the understanding
of a plasmonic system.
The most common method to map these iso-frequency contours is the use of reflec-
tivity plots for a range of azimuthal angle,
φ
. By collecting reflectivity as a function of
polar angle
θ
for a range of
φ
, the position of the SPP mode in reciprocal space may
be mapped by the transformation
R
(
θ, φ
)
R
(
k
0
sin θ, φ
)
R
(
k
x
, k
y
). This method
has several drawbacks. The accuracy of determining a SPP mode position in θ can be
limited if the plane of incidence lies parallel to a SPP contour, as the reflectivity minima
associated with the SPP will be broad. The resolution of this method is also limited by
the step size in
φ
, which, for experimental purposes, may have to be limited to prevent
the experiment taking a prohibitively long time (particularly with silver samples, which
degrade over time).
Several methods to measure the reciprocal space map for plasmonic systems directly
have been previously published. Reports of using crystal defects [
104
] to couple to
the SPP modes have revealed the general shape of the underlying lattice of photonic
crystals. Direct recording of the ‘photonic Fermi surfaces’ using a plasmon topography
technique [
105
], which images a sample in the Fourier plane of an optical microscope
have been reported, as well as direct imaging of the contours using scattering through
transmission gratings [
106
]. Direct observation of dispersion in a single plane of incidence
using multiple wavelengths simultaneously is also found in the literature [
107
109
], but
these methods are not related to mapping the full SPP contours in k-space but rather
obtaining the in-plane dispersion curve directly.
In this thesis, we present a new method for directly obtaining the
k
-space contours
51
4. Experimental Methodology
of SPP at single frequencies by adapting an imaging scatterometer used commonly
in Natural Photonics investigations. This new method records the SPP contours of a
reflection grating across the entire light cone to a high degree of accuracy, and the results
can be numerically compared to theoretical predictions. By collating the contours from
a range of frequencies, the entire 3D band structure of the SPPs may be experimentally
determined.
4.3.1 Scatterometry
Scatterometry has been used previously to record the scattering patterns from samples
found in Natural Photonics, such as the naturally diffuse scattering of the Chrysochroa
fulgidissima beetle [
110
] or the Morpho aega butterfly [
111
]. It has also been used to
show the hexagonal shaped Brilloun zone of a pseudo-hexagonal lattice found in the E.
imperialis moth[112].
In this work, the previously reported experimental arrangement [
111
] has been
modified by the addition of spectral filters to limit the image acquired to a single
wavelength. Using a plasmonic grating sample, the scatterometer directly images the
iso-frequency surface of the plasmonic dispersion. By applying a simple deformation to
the image (also an adaptation new in this thesis) the scatterometer provides a map of
the coupled SPP iso-frequency contours in momentum-space over the entire incident
light cone.
4.3.2 Principle of Operation
A schematic of the scatterometry arrangement is shown in figure 4.6. White light
is directed through a collimating lens, L1 and an optional linear polariser allows the
investigation of the polarisation sensitivity of the acquired image. The white light is then
focussed through an alignment pinhole and the beam is reflected via the beam-splitter,
BS, on to an ellipsoidal mirror with an eccentricity of 0.833. The mirror, M1, focusses
the light onto the sample, positioned at G.
If aligned precisely, the cone of light focussed by the mirror will include light from
all directions (all azimuthal,
φ
, angles) and a range of polar angle ranging from
θ
2
to
θ
90
. The lower limit of
θ
is determined by the shadow cast by the sample itself.
The reflected light from the sample is then collected by the same ellipsoidal mirror,
M1 and is focussed through a second alignment pinhole, P2, positioned at the mirror’s
secondary focal point. This light is then collimated which causes the polar angle
θ
to be
approximately linearly proportional to the obtained image’s radial axis
. The beam is
The small correction applied to the image that is required to obtain the true
R
(
θ, φ
) image is
detailed later in section 4.3.5
52
4. Experimental Methodology
L1
Pol
L2
P1
BS
M1
P2
Pol
G
L3 F L4
CCD
r
θ
Figure 4.6: Experimental arrangement for the modified imaging scatterometry system.
The angle
θ
of the light impinging on the grating, G, is approximatly linearly proportional
to the image’s radial axis, r.
finally passed through a spectral filter and imaged using a CCD camera. The acquired
image is a directly mapped reflectivity plot of all polar and azimuthal angles,
R
(
θ, φ
),
with a high resolution determined by the pixel size and density of the CCD. Using a
range of spectral filters, the wavelength, and hence energy, of the map can be selected.
The filters used are: 450
±
5 nm, 500
±
5 nm, 550
±
5 nm, 580
±
5 nm, 600
±
5 nm,
650
±
5 nm, 700
±
5 nm and 750
±
5 nm. The reflectivity of the sample for a single
wavelength over the range 0
< φ <
90
and 2
< θ <
90
is thus recorded in a single
image by the CCD.
Light of appropriate in-plane momentum may couple to diffracted SPP contours
that exist within the recorded zero-order light circle. At the values of
θ
and
φ
which
match the momentum of such a SPP, a reflectivity anomaly will be found in the acquired
image. These reflectivity extrema, to a first approximation, map the position of the
SPP iso-frequency contours over the large range of
θ
and
φ
available to this experiment.
4.3.3 Sample Preparation
Since the lower limit of polar angle,
θ
is determined by the size of the sample area, a
small sample area is desirable to maximise the polar angle range. The grating samples
are prepared by template stripping from a Si master, (a process detailed in section 4.1)
to the centre of a large plate of glass, which acts as the substrate. The grating is then
further reduced in size by using a clean razor blade to remove excess sample area.
Most previously reported work using scatterometry use a glass pipette to place the
sample at the focal point of the ellipsoidal mirror. The pipette casts a characteristic
53
4. Experimental Methodology
Figure 4.7: Four raw images from the scatterometer for a wavelength of
λ
0
= 650 nm.
The sample is a rectangular bi-grating supporting SPPs, as will be explored in chapter 5.
The polarisation states are (left to right) (1) polarisation
a
, which gives good contrast of
the thin dark SPP bands, (2) crossed polarisers with the first polariser set to polarisation
a
and the second set at 90
to this, named polarisation
b
(3) the alternative crossed
polarisation, with the first polariser set to
b
and the second polariser set to polarisation
a, and (4) The final polarisation, with both polarisers set to b.
shadow that may hide important spatial features. By using a large glass plate substrate
that covers the entirety of the the mirror face, shadows from alternative positioning
devices are avoided. It is found, by comparison of our final images with calculated
diffraction edges and theoretical predictions, that any distortion effects due to refraction
through the substrate are negligible.
4.3.4 The Role of Polarisation
The use of unpolarised light in this experiment is often adequate for the complete
mapping of SPP modes diffracted into the zero-order light circle. Greater contrast of
particular modes may be achieved by the use of two linear polarisers placed in the set-up
at the positions marked. The complicated polarisation arrangement, due to the focussing
from the ellipsoidal mirror, can cause ‘lobes’ of high and low reflectivity to appear, as the
multiple reflections on the mirror lead to polarisation conversion for all but two radial
axes. Still, if a single contour is of particular interest, the experimenter may adjust
the polarisers to obtain the greatest contrast between the reflectivity extrema and the
background. Depending on the polarisation states of the two polarisers, SPPs interacting
with light may present as a maximum, minimum or inflection of the reflectivity. This is
shown in an example dataset in figure 4.7.
These scattergrams are for a rectangular bigrating which will be discussed in chapter
54
4. Experimental Methodology
5. Notice that in the first polarisation case (left most), rather flat SPP contours are
seen clearly as dark contours crossing at the center (θ = 0
), while for the second case
when the polarisers are crossed, the same SPPs exhibit as bright bands. This is a
demonstration of SPP mediated polarisation conversion[
34
]. Diffracted SPP contours
due to a second orthogonal pitch are best seen in the last polarisation case, and in this
case have a far smaller radius of curvature. Discussion of these particular experimental
results will continue in chapter 5.
The polarisation is chosen in each scattergram case depending on which SPP modes
are to be best observed, and this choice will be stated along with the results.
Notice in these scattergrams how 4 distinct dark lobes in the first polarisation case
change to bright lobes in the second and third cases, and return to dark in the last
case. These are regions of polarisation conversion due to the mirror, and not due to the
sample.
4.3.5 Corrections and Momentum-Space Deformation
To obtain a map of momentum-space from the scattergrams, two image corrections are
required. The first corrects the aberration of the ellipsoidal mirror to obtain an image
whose radial axis is linearly scaled with respect to the polar angle,
θ
. This correction is
detailed in previous work [111].
The second deformation, scales the radial axis of the image to be proportional to
the in-plane momentum, such that the reflectivity plot, R(θ, φ) becomes,
R(θ, φ) R(k
0
sin θ, φ) R(k
x
, k
y
)
Where
k
0
sin θ
is the in-plane momentum for a specular reflected photon in the plane
of incidence, at an azimuthal angle
φ
. The effect of this deformation is illustrated in
figure 4.8. This figure illustrates an important experimental consideration for the use
of scatterometry for the direct imaging of momentum space. Due to the
sin θ
radial
dependence of the final
k
-space diagram, higher values of
θ
provide less information
than the angles close to normal incidence. It is then possible to map a very large area
of momentum space, while only illuminating up to θ = 70
80
degrees.
4.3.6 Example Dataset
The processing of a raw dataset is shown in figure 4.9. The example shows a diffraction
grating with a period of 600 nm illuminated at a wavelength of
λ
0
= 650 ± 5 nm
. In
figure 4.9(a), the raw image taken from the scatterometer is shown, with two dark
contours of reflectivity minima showing the position of the
±k
gx
scattered SPP contours
55
4. Experimental Methodology
θ
0
45
90
θ
0 90
R(θ, φ)
(a)
k
x
k
y
0
15
30
45
60
90
θ
k
0
0 k
0
k
0
0 k
0
R(k
0
sin θ, φ)
(b)
Figure 4.8: The image correction for
R
(
θ, φ
)
R
(
k
0
sin θ, φ
). The black circles on both
diagrams show contours of equal angle, ranging from 10
to 90
in 10
steps.
in
θ φ
space. The image correction is then applied and the result is shown in figure
4.9(b), producing a momentum-space picture. Notice how the SPP contours have become
broader and towards the radial edge curve inwards to form a more circle-like dispersion
than they do in the raw image. Figure 4.9(c) shows the data adjusted for an azimuthal
angle of rotation so that the SPP contours lie in the (arbitrarily chosen)
k
x
direction.
This is to ensure our definitions between
k
x
and
k
gx
remain consistent. Two blue lines
are added to figure 4.9(c), which are the calculated positions of the
±k
gx
diffracted
light cones. The SPP contours, as expected, lie just outside these cones (see background
theory chapter 2). The contours sit closer to the diffraction lines along
k
x
= 0 than they
do at the extremes of the radius of the dataset, due to a small anisotropic dispersion of
the SPPs. This anisotropic dispersion is because this example diffraction grating is, in
fact, a complex grating type called a zigzag grating, which will be explained in detail in
chapters 7 and 8.
56
4. Experimental Methodology
(a) Raw data (b) deformation
(c) φ correction
Figure 4.9: The processing of a raw dataset from the scatterometer at a wavelength
of
λ
0
= 650 nm. (a) Raw image obtained from scatterometer, (b) the image after the
applied momentum-space deformation, and (c) the final image with corrected azimuthal
angle and addition of calculated diffraction circles.
57
Chapter 5
Optical Response of Metallic
Rectangular Bigratings
5.1 Introduction
Bigratings consist of two monogratings ‘crossed’ at some angle relative to each other [
93
].
The study of SPPs on such gratings has covered various symmetries, and various novel
optical effects have been observed. Hole arrays that exhibit extraordinary enhanced
transmission that is mediated by SPPs travelling along the surface [
62
,
79
,
113
] are types
of bigrating, often with square symmetry. Full photonic band gaps for surface waves
has been demonstrated on hexagonal symmetry gratings [
114
], as has total absorption
of unpolarised light [77], and absorption of light across a broad angle range [115].
This chapter details the optical response of metal bigratings with rectangular lattice
symmetry. This symmetry of gratings has received comparatively less attention in the
published literature and serves as a good introduction to the methods and physics of SPP
on bigratings observed throughout this thesis. This chapter also shows experimentally
how anisotropic propagation of SPPs may be designed by controlling the scattering
amplitude in one direction and leaving the orthogonal direction unaltered.
The gratings used for this investigation, and all subsequent investigations in this
thesis, have straight-walled grooves and are commonly referred to as lamellar gratings.
It is instructive, then, to review the scattering properties of such lamellar type gratings,
and so a simple Fourier analysis giving the scattering amplitudes of a lamellar grating
is presented in section 5.3.1.
A fabricated rectangular bigrating is used to demonstrate the coupling to, and
observation of SPPs supported by such a structure. The reflectivity of a silver rectangular
bigrating is then used to experimentally map the SPP dispersion in section 5.3.2. The
observed coupling of light and the interaction of the SPP modes is explained with
58
5. Optical Response of Metallic Rectangular Bigratings
respect to the available scattering harmonics present in the constituent gratings. These
scattering harmonics are inferred from the Fourier expansion found in section 5.3.1 and
SEMs of the sample. These results serve to demonstrate the experimental techniques
used to map the SPP dispersion and also how the mark-to-space ratio is a key parameter
in the understanding of the scattering on such lamellar gratings.
The observation of polarisation conversion is also experimentally studied in section
5.4, showing that the reflected polarisation may be rotated when the plane of incidence
is not along an axis of high symmetry.
The new technique of imaging scatterometry is used in section 5.3.3 to map the iso-
frequency SPP contours in reciprocal space for the grating. The obtained iso-frequency
maps show the formation of a band-gap at the first BZ boundary, a result that is
reproduced using a FEM model and shows good agreement. This model is used to
calculate the electric field of the two different SPP standing waves that occur at the BZ
boundary.
Finally, section 5.5 of this chapter shows how the deepening of one of the constituent
gratings provides a mechanism to control the anisotropic SPP propagation along the
surface. The deformation of the SPP iso-frequency contours is experimentally recorded
using imaging scatterometry.
5.2 The Rectangular Bigrating
A rectangular bigrating consists of two monogratings of different pitches ‘crossed’ at an
angle
α
= 90
. The coordinate system for this type of grating is shown in figure 5.1.
The plane of incidence is defined at an azimuthal angle of
φ
= 0
when the wavevector of
incidence light, impinging at some polar angle,
θ
, lies in the
xz
plane. When the electric
field vector of the impinging radiation is contained within the plane of incidence, the light
is said to be TM polarised, and when the electric vector lies orthogonal to the plane, it is
TE polarised. The
x
-direction is in the plane of diffraction for the longer-pitch grating,
which possesses a periodicity of
λ
gx
, and for all the gratings presented in this chapter is
equal to 600 nm. The grating vector for this grating is defined as
k
gx
= 2
π
ˆ
x
gx
. The
second, shorter-pitch grating lies at an angle of
α
= 90
to the
x
-direction, and has
λ
gy
= 400 nm in this chapter. As before, the grating vector of this short-pitch grating is
defined as
k
gy
= 2
π
ˆ
y
gy
. The depths of the gratings are
d
1
and
d
2
, and are chosen in
this chapter so that d
1
= d
2
40 nm.
59
5. Optical Response of Metallic Rectangular Bigratings
φ
θ
E
T M
E
T E
d
1
x
y
z
λ
gx
λ
gy
plane of incidence
d
2
Figure 5.1: The coordinate system for a rectangular grating. Light is incident on the
surface at a polar angle
θ
and an azimuthal angle
φ
. The plane of incidence contains
the wavevector of the light, and the polarisations are defined accordingly.
5.3 Dispersion of SPPs on Rectangular Bigratings
5.3.1 Scattering Components on Lamellar Gratings
In this chapter we use rectangular bigratings that have been fabricated using the electron
beam lithography and template stripping method outline in chapter 4. These have
surface relief grooves whose groove profiles are to a good approximation represented
by a rectangular step function. Gratings such as this are often referred to as ‘lamellar’
gratings, ‘binary’ gratings, or occasionally ‘rectangular gratings’. To avoid confusion
in this chapter, we shall reserve the label ‘rectangular’ for discussion of the bigrating
lattice, and refer to the groove profile shape as simply ‘lamellar’.
To explain the SPP coupling and dispersion on rectangular bigratings we must gain
a qualitative understanding of the scattering strength from lamellar profile gratings.
The strength of scattering for fields at the grating surface is determined by the Fourier
components of the surface profile. This is because the incident field’s wavevector will
be modified by the addition or subtraction of an integer number of surface profile
wavevectors, and the diffraction efficiency into each order is proportional to the Fourier
coefficients of these harmonics squared [
116
,
117
]. To fully understand the excitation
and interaction of SPPs on rectangular bigratings, we must first obtain expressions for
60
5. Optical Response of Metallic Rectangular Bigratings
these Fourier harmonics present for lamellar gratings.
A step function representing a monograting surface profile can be expressed by the
piecewise linear function,
f(x) =
A, L < x <
mL
2
A,
mL
2
< x <
mL
2
A,
mL
2
< x < L
(5.1)
Where
A
is the amplitude of the grating,
λ
g
= 2
L
is the period of the periodic function,
and
m
is a variable that affects the groove widths. Two common ways of defining the
relative size of the region of grooves to the peaks for such functions are the ‘mark-to-space
ratio’ (
MSR
) and the ‘duty cycle’ or ‘fill-fraction (Γ) of the function. The MSR is
defined as the ratio between the peak length and the trough length, while the duty
cycle is defined as the fraction of the period which the peak occupies. The variable
m
in
the piecewise linear function was used for the convenience of calculation for the Fourier
series. MSR and Γ are defined in terms of m as,
MSR =
m
2 m
Γ =
m
2
MSR varies between 0 < M SR < while Γ varies between 0 < Γ < 1.
Several examples of the grating function (equation 5.1) are shown in figure 5.2(a).
This function is an even function and so may be expressed as a Fourier sum containing
only cosine terms, with coefficients a
n
given by,
a
n
=
4A
sin
m
2
(5.2)
The Fourier sum is then,
F (x) =
X
n=1
4A
sin
m
2
cos
x
L
The amplitudes of the first four Fourier coefficients as a function of the duty
cycle of the grating is shown in figure 5.2(b). The gratings fabricated in this chapter
were designed for a Γ = 0
.
5, which gives peaks equal in length to the troughs. For
Γ = 0
.
5
,
(
MSR
= 1), there are no even Fourier components present in the surface
expansion (
a
2,4,...
= 0). This means that, for a lamellar grating with Γ = 0
.
5, there is no
direct scattering of light by even scattering vectors, and the even-ordered diffraction is
61
5. Optical Response of Metallic Rectangular Bigratings
x
f(A, Γ)
-L 0 L
-A 0 A
(a)
0.0 0.5 1.0 1.5
Duty Cycle, Γ
|a
n
|
2
0.00 0.25 0.50 0.75 1.00
n = 1
n = 2
n = 3
n = 4
(b)
Figure 5.2: (a) Three examples of unit cells for lamellar grating profiles represented
by equation 5.1.
f
(
A,
Γ) for each case is
f
(
A,
0
.
5),
f
(0
.
7
A,
0
.
75) and
f
(0
.
5
A,
0
.
25). (b)
Square of the Fourier coefficients,
a
n
, defined by equation 5.2, in lamellar groove profiles
as a function of Γ for
n
= 1
,
2
,
3
,
4. The two grey vertical lines highlight the two values
of Γ measured from the experimental sample used in section 5.3.2.
consequently very weak. Even-ordered diffraction in this case is not forbidden, however,
as light may scatter multiple times via odd-ordered scattering events (clearly present for
Γ = 0
.
5 in figure 5.2(b)), but the strength of this multiple scattering process is normally
quite weak. Small fabrication errors in Γ will lead to non-zero values of even Fourier
harmonics, and in this case direct scattering events will occur.
The expression of the Fourier components as a function of Γ is relevant to all the
gratings produced and measured in this thesis.
5.3.2 Experimental Mapping of SPP Dispersion
This section demonstrates the experimental mapping of SPP dispersion on rectangular
gratings using reflectivity measurements. An electron micrograph of the fabricated
grating is shown in figure 5.3.
This sample is a silver grating fabricated using electron beam lithography to produce
a silicon master, followed by the coating of the master with silver and then template
stripped on to a clean glass substrate. This process is detailed in depth in chapter 4.
The measured parameters from the SEM were
λ
gx
= 592
±
10 nm and
λ
gy
= 395
±
9 nm
with the duty cycle of the grating in the
x
direction measured as Γ
x
= 0
.
45 and in the
y
direction as Γ
y
= 0
.
54. The variation of Γ from the designed duty cycle of Γ = 0
.
5
will lead to possible direct scattering events for both odd and even grating harmonics,
62
5. Optical Response of Metallic Rectangular Bigratings
(a) (b)
Figure 5.3: (a) A scanning electron micrograph of the template-stripped rectangular
bigrating in silver. (b) A higher magnification of the surface showing small amounts of
surface roughness attributed to anistropic etching of the Si master. The parameters are
λ
x
= 600 nm, λ
y
= 400 nm.
as can be seen by referring back to figure 5.2(b), in which these two values of Γ are
marked with grey lines.
The dispersion of SPPs on such a grating was experimentally mapped from the
sample’s zero-order reflectivity using the technique outlined in chapter 4. Using this
method, the re-radiated light from SPPs will in general differ in phase to the zero-order
reflection and so produce reflectivity features that map the SPP dispersion. The polar
angle range used was 7
< θ <
60
in steps of 0
.
5
, and the wavelength range used was
400 nm
< λ
0
<
800 nm in steps of 2 nm. The resultant reflectivity plots for different
polarisations and azimuthal angles are shown in figure 5.4. Since both periodicities on
this bigrating will provide observable scattering at some point in our discussions, we
adopt the notation of using two integers (
m, n
) to represent the
mk
xg
+
nk
yg
scattering
events.
Figure 5.4(a) shows the measured reflectivity of TM polarised light for a rectangular
grating at
φ
= 0
. The blue lines show the momentum states of grazing photons for
various diffracted orders and in this figure include the
±
1
k
gx
, +2
k
gx
scattered and the
zero-order light line. No diffraction lines associated for the
λ
gy
period are present, as
the pitch of
λ
gy
was chosen to ensure, for the orientation of the grating at
φ
= 0
, there
was no diffraction present from the shorter period in this frequency range.
The figure shows dark bands of low reflectivity that map the positions of the
±
1
k
gx
and +2
k
gx
scattered SPP modes. These modes lie beyond their respective diffracted
63
5. Optical Response of Metallic Rectangular Bigratings
in-plane wavevector, k
x
(m
1
)
angular frequency, ω (rad s
1
)
0.0
0.2
0.4
0.6
0.8
1.0
0.2 0.4 0.6 0.8 1 1.2
× 10
7
2.5 3 3.5 4
× 10
15
(-1,0)
(1,0)
(2,0)
(0,0)
(a) TM at φ = 0
in-plane wavevector, k
y
(m
1
)
angular frequency, ω (rad s
1
)
0.0
0.2
0.4
0.6
0.8
1.0
0.2 0.4 0.6 0.8 1 1.2
× 10
7
2.5 3 3.5 4
× 10
15
(1,0)
(1,1)
(0,1)
(0,0)
(b) TE at φ = 90
in-plane wavevector, k
x
(m
1
)
angular frequency, ω (rad s
1
)
0.0
0.2
0.4
0.6
0.8
1.0
0.2 0.4 0.6 0.8 1 1.2
× 10
7
2.5 3 3.5 4
× 10
15
(-1,0)
(1,0)
(2,0)
(0,0)
(c) TE at φ = 0
in-plane wavevector, k
y
(m
1
)
angular frequency, ω (rad s
1
)
0.0
0.2
0.4
0.6
0.8
1.0
0.2 0.4 0.6 0.8 1 1.2
× 10
7
2.5 3 3.5 4
× 10
15
(1,0)
(1,1)
(0,1)
(0,0)
(d) TM at φ = 90
Figure 5.4: The reflectivity for different polarisations and azimuthal angles for a
rectangular grating mapped as a function of (
ω, k
) in the case of (a) TM polarised light
at
φ
= 0
, (b) TE polarised light at
φ
= 90
, (c) TE polarised light at
φ
= 0
and (d)
TM polarised light at φ = 90
.
light lines, as the momentum of SPPs is greater than that of corresponding scattered
light. Several conclusions can be made about this grating from the observation of the
SPPs. Firstly, the
±
1
k
gx
scattered SPP is coupled strongly to light, presenting as
a reflectivity minimum with only
20% of the light being reflected. This is due to
the phase of the re-radiated light from the SPPs being out-of-phase with the direct
(non-interacting) reflected light from the surface. The +2
k
gx
SPP is also coupled well
and is observed as a dark band. This suggests that there is a 2
k
gx
component in the
grating surface profile, as for light to interact with a +2
k
gx
scattered SPP and present
as a dark band the light has undergone a single scattering process equal to +2
k
gx
,
64
5. Optical Response of Metallic Rectangular Bigratings
which is only strongly observed if the profile itself contains a 2
k
gx
component. This is
expected for Γ
x
= 0
.
45, as measured from the SEM in figure 5.3. If Γ
x
= 0
.
5, there
would be no 2
k
gx
component in the grating surface and the +2
k
gx
could only present
in the reflectivity data via weak multiple scattering processes.
At the first BZ boundary these two counter-propagating modes meet and form
a small band-gap, which is not fully mapped as it occurs at the upper limit of the
experiment’s frequency range. The observable part of this band-gap is the lower band
edge occurring at
k
x
=
π
gx
= 0
.
52
×
10
7
m
1
. These SPPs coupling together to form a
band-gap means that the grating profile also contains a 3
k
gx
component which couples
these two modes together (see chapter 2). A 3
k
gx
component in the surface profile is
fully expected for lamellar gratings with Γ
x
= 0.45.
Figure 5.4(c) shows the same reflectivity map, this time for TE polarised light. TE
polarised light provides no normal component of electric field to the surface of the long
pitch grating, and so no SPPs are excited. In the top corner of the dispersion, at larger
values of
k
x
and at high frequencies, there is a broad dark region that is attributed to a
largely over-coupled mode associated with the
nk
gy
scattered SPPs, for which the TE
polarisation is suitable for SPP excitation.
Rotating the grating by
φ
= 90
, so that the plane of incidence now lies along the
k
y
direction allows the observation of the
±
1
k
gx
and
±
2
k
gx
SPPs that present now as
out-of-plane modes when the light is TE polarised (Figure 5.4(b)). These mapped SPP
bands follow curved diffraction lines formed by the conic intersection of an out-of-plane
diffracted light cone (see the discussion in chapter 2).
A band-gap is observed in this direction between the (
±
1
,
0) and the (
±
1
,
1) scattered
SPPs, with the centre of the band-gap at
ω
= 3
.
45
×
10
15
rad s
1
. This corresponds to
an illumination wavelength of
λ
0
= 546 nm, and we shall use a wavelength close to this
to observe this band-gap using scatterometry later in section 5.3.3.
These bands are coupled to well beneath these diffraction lines, and appear weakly
coupled once the band passes above the diffraction lines. This is due to the re-radiated
light from SPPs now being able to couple not just to the zero-order light, but also to
the allowed modes of diffracted light scattering along the
k
x
axis. This additional loss
mechanism of the light leaves less of the out-of-phase re-radiated light in the measured
specular reflection.
The straight lines in figure 5.4(b) show the diffraction lines associated with the
+1
k
gy
light and the zero-order light line. No SPP is observed that is associated with
the +1
k
gy
light line, as the polarisation conditions are not met for the excitation of this
mode.
The final dispersion map is shown in 5.4(d) for
φ
= 90
and TM polarised light. In this
case the SPPs associated with the
nk
gy
scattering events should be observed. Following
65
5. Optical Response of Metallic Rectangular Bigratings
(a) (b)
Figure 5.5: (a) A scattergram of a rectangular grating mapped to
k
-space for an
illumination wavelength of
λ
0
= 550 nm. The red square indicates the position of the BZ
boundary. The white region bounded in the scattergram is replicated in (b), modelled
using the FEM.
the +1
k
gy
light line, a dispersive mode, broader than the SPP bands seen previously,
is observed. This mode forms a large band-gap at
k
y
= 0, with a forbidden frequency
range observed between
3.2 × 10
15
rad s
1
< ω < 3.5 × 10
15
rad s
1
, suggesting a large
2
k
gy
component in the grating which may directly couple the counter-propagating SPPs
together. Above this band-gap, a very large broad absorption band is seen, with the
reflectivity being reduced further inside the +1
k
gy
diffraction cone. This is attributed
to largely over-coupled SPP modes, and the extra available loss channels inside the
diffraction cone. The small dark band observed at
ω 3.1 × 10
15
rad s
1
is the lower
band edge of a band gap formed at the 1
st
BZ boundary in the k
y
direction.
5.3.3 Band Gap Observation Using Scatterometry
Scatterometry has been used in this section to observe the band-gaps that form at the
first BZ in the
k
y
direction due to the interaction of a +1
k
gx
and a
1
k
gx
scattered
SPP. This is a new technique presented in this thesis, and the methodology is covered in
chapter 4. Figure 5.5(a) shows the results for a rectangular grating for an illuminating
wavelength of λ
0
= 550 nm.
In this figure, the BZ boundary is shown as a red rectangle and the
±
1
k
gx
scattered
SPPs present as dark bands in the reflectivity. The
±
1
k
gy
scattered SPPs are also
observed as extremely weak bright bands, due to the selected polarisation for this
scattergram, which was chosen to provide greatest contrast for the
±
1
k
gx
scattered
66
5. Optical Response of Metallic Rectangular Bigratings
BZ
(1, 0) (1, 0)
k
y
k
x
(a)
BZ
(1, 0)
(1, 0)
k
y
k
x
(1, 1)
(1, 1)
(b)
Figure 5.6: (a) Cartoon of the (
1
,
0) and (1
,
0) SPP iso-frequency contours at the
BZ boundary, from the original continuous SPP contours (dotted lines) to the split
modes (solid lines). (b) By considering the other interactions occurring at this point (or
by using the mirror symmetry of the BZ boundary), the continuous SPP contours are
recovered, leading to the observation of the SPP contour shape measured in figure 5.5.
SPPs. The
±
2
k
gx
scattered SPPs are also observed as dark bands near the edges of
the plot at
|k
x
|
= 0
.
9
×
10
7
m
1
. This sample was produced from the same master as
the grating presented in section 5.3.2, so a 2
k
gx
component in the surface profile is not
unexpected.
As the scattered SPP contours cross at the BZ boundary, they are seen to split.
The contours intersect the BZ boundary perpendicularly, indicating that there is zero
group velocity in the
k
y
direction, as one would expect for a set of standing waves. The
shape of these contours is explained in figure 5.6 as the splitting of the modes due to a
band-gap.
The region of the band-gap is highlighted in white in figure 5.5(a), and in figure
5.5(b) the data in the area shown has been reproduced using a FEM model. The SPP
contour at point A appears discontinuous at this boundary, but this cannot be true. The
SPP contours are never discontinuous due to the translational symmetry of reciprocal
space as illustrated in figure 5.6, and this apparent discontinuity in the contour is in
fact the SPP simply being poorly coupled beyond the point of the BZ boundary. The
SPP contour at A will connect with the SPP eigenmode associated with the (
1
,
1)
scattered SPP contour, and the coupling of light to this eigenmode is sufficiently weak
as to cause this apparent discontinuity.
The model shows good agreement to the obtained scattergram with the parameters
as follows:
d
1
=
d
2
= 35 nm
, λ
gx
= 600 nm
, λ
gy
= 400 nm with Γ
x
= 0
.
45, Γ
y
= 0
.
54
and silver parameters from literature [
118
]. From this model, the electric field profiles
67
5. Optical Response of Metallic Rectangular Bigratings
(a) High-energy solution (b) Low-energy solution
Figure 5.7: Colour plots showing the magnitude of electric field at a set temporal phase
(colour scale) and electric vector direction (arrows) for the (a) high energy and (b) low
energy SPP standing waves occurring at the BZ boundary.
in the
yz
plane were extracted and plotted in figure 5.7. These band gap solutions
correspond to the points A and B in figure 5.5(b), where the SPP contour intersects the
BZ boundary perpendicularly, indicating the SPP mode possesses zero group velocity
in the k
y
direction, indicative of a standing wave.
The electric field plots show that both standing waves have a wavevector such that
k
y
=
k
gy
/
2, as is expected at the first BZ boundary. One mode extends further into the
air, and is the ‘photon-like’, high energy solution (figure 5.7(a)). This shows that charge
accumulates mostly on the grating peaks, while the charge is localised to the grating
grooves in the lower energy case of figure 5.7(b), in agreement with the analogous case
of simple sinousoidal gratings discussed in chapter 2.
5.4 Polarisation Conversion on Rectangular Symmetry
The conversion of the polarisation of light is of some importance in many plasmonic
applications. The two mechanisms of polarisation conversion outlined in chapter 2 detail
how the geometry of a diffraction grating and the presence of SPPs on a surface may
mediate significant changes in the polarisation state of the reflected light. Both these
mediating processes are possible when the light impinging on a grating surface does so
along an axis of reduced symmetry. The basic principle of this is quite intuitive; one
68
5. Optical Response of Metallic Rectangular Bigratings
does not expect any physical response of the system to possess a different symmetry
to that with which it started. So, when the electric field of an incident light beam lies
in a plane of mirror symmetry shared by both the light and a surface, it is correctly
expected that the reflected light’s electric field will occupy the same plane. This offers
us an intuitive understanding for a typical diffraction grating with a single periodicity
reflecting polarised light. We do not expect the polarisation of the TE or TM polarised
light to change upon reflection when the electric field lies either parallel or perpendicular
to the grating’s grooves as this is when the polarised electric (and magnetic) field vectors
lie in the grating’s mirror symmetry planes (when
φ
= 0
or
φ
= 90
). However, if the
light field vectors do not lie in the mirror symmetry planes of the diffraction grating,
say by rotating the grating azimuthally by an angle which is not
φ
= 0
or
φ
= 90
, the
mirror symmetry considerations place no constraints on the rotation of polarisation,
and polarisation conversion may be observed. In fact, maximum polarisation conversion
of the incident light (for a monograting) occurs half way between these two angles, at
φ = 45
This is a general symmetry description of polarisation conversion, which gives us
an intuitive insight as to when we might expect the rotation of light’s polarisation
to change, or when it will not. On a square bigrating, with
λ
gx
=
λ
gy
, there is an
additional plane of mirror symmetry, at
φ
= 45
. At this angle, again, no polarisation
conversion is observed and the polarisation state of reflected light is unchanged [119].
In the rectangular bigrating case, the plane of incidence will only be collinear with
planes of mirror symmetry when
φ
= 0
,
90
,
180
,
270
. We expect to observe rotation
of the polarisation state of incident light for any azimuthal angle which is not
φ
= 0
or
φ
= 90
. To highlight the differences in the rectangular grating to the square grating,
we choose here an angle of
φ
= 45
to investigate the grating’s ability to rotate the
polarisation.
To record the polarisation conversion of light from TE to TM (and from TM to
TE), spectra were recorded in the range 450 nm
< λ
0
<
800 nm in steps of 2 nm.
The incident light was passed first through a linear polariser set to either TE or TM,
and then directed on to the grating surface, which was oriented at
φ
= 45
. The
zero-order reflected light was then passed through a second linear polariser, which for
the purposes of detecting polarisation conversion was crossed at 90
with respect to
the first polariser. Normalisation of the polarisation converted signal was performed by
dividing the reflectivity spectra with a straight-through spectra of linearly polarised
light (both polarisers set to the same polarisation).
Figure 5.8 shows the results of this experiment. The upper half of the figure shows
the reflectivity for uncrossed (black) polarisers both set to the TE polarisation, and the
and crossed (red) polarisers set to TE and TM respectively. Resonant features exist
69
5. Optical Response of Metallic Rectangular Bigratings
0.0 0.4 0.8
R
450 500 550 600 650 700 750 800
0.5 0.7 0.9
rss45x[rss45y == angle]
R
T E:T E
+ R
T E:T M
wavelength, λ
0
(nm)
Figure 5.8: Spectra of polarisation conversion on the rectangular grating at
φ
= 45
. Top
panel: The reflection of TE polarised light,
R
T E:T E
(black) and polarisation conversion
R
T E:T M
(red). Lower panel: The addition of these two reflectance curves gives a more
precise position of the SPP modes.
in both spectra characteristic of SPP interaction with the light. For the polarisation
conversion signal (red line) we see a large broad region of polarisation conversion
between
500 nm < λ
0
< 600 nm
, which we attribute to polarisation conversion mediated
by largely over-coupled modes at high frequencies, evidence for which we observed
previously in section 5.3.2. The second resonant feature at
λ
0
690 nm presents as an
inflection of the polarisation converted signal and is in the same region as the reflectivity
minimum in the R
T E
case (black line).
This highlights an important point when dealing with the mapping of SPP position in
systems of low symmetry. Normally, the measurement of the SPP energy or momentum
is accomplished by measuring the resonant minima (or maxima) position in energy or
momentum space. However, when SPPs propagate along axes of broken symmetry, the
phase of the re-radiated light into a single polarisation state is insufficient to determine
the exact SPP location, as additional loss channels are available (The SPP may couple
out to a different polarisation state, and the phase of this light with respect to the
70
5. Optical Response of Metallic Rectangular Bigratings
incident field may be different). This is highlighted in the lower portion of figure 5.8,
where the two obtained reflectivities have been summed to obtain the reflectivity of
both TE:TE and TE:TM polarised reflections. The result shows reflectivity resonance
minima which better position the SPP in wavelength, as some of the additional phases
have been accounted for through interference by the summation of the two reflectivities.
The grey lines across both figures show that in the original upper spectra, the mode
positions do not correspond to this corrected (bottom) spectra.
5.5 Controlling SPP Anisotropy Using Rectangular Bi-
gratings
It was demonstrated in section 5.3.2 that SPP band-gaps form in both the
k
gx
and
k
gy
directions on a rectangular bigrating. The size of these band-gaps is proportional to the
diffraction efficiency required to couple two counter-propagating SPPs together to form
the required SPP standing waves. This diffraction efficiency into a particular diffracted
order is also proportional to the Fourier components squared, and an expression for
these was derived in section 5.3.1.
Examining equation 5.2, it is possible to increase the diffraction efficiencies of a
constituent grating by simply increasing the grating’s amplitude. Since on a lamellar
type grating, each Fourier component is proportional to this grating amplitude, the size
of all the possible SPP band-gaps may be controlled by simply deepening the grating.
On a rectangular bigrating there are two constituent lamellar gratings oriented
orthogonal to one another. SPP band-gaps in one direction may be controlled by
deepening the appropriate grating, leaving the band-gaps that form in the orthogonal
direction along the other constituent grating largely unaffected. This should allow the
design of SPP anisotropy, where the SPP dispersion varies largely depending on the
direction in which the SPP travels along the grating.
In this section, we deepen the short-pitch grating and observe the effect of increased
SPP anisotropy using imaging scatterometery. The variable we adjust is
d
2
, and the
effect on the surface profile of increasing this is shown in figure 5.9.
Two bigratings were produced for this experiment, with the aim that they be
identical save for the depth of the shorter-pitch grating,
d
2
. The two gratings are
manufactured at the same time via electron beam lithography, using the same electron
dose to expose the pattern. The long pitch is
λ
gx
= 600 nm and for the short pitch
λ
gy
= 400 nm. The target depth of the long pitch for both gratings was
d
= 40 nm,
which is achieved by reactive ion etching the masters in the same etching exposure run.
The second short-pitch grating was then added to the master and the depth varied
71
5. Optical Response of Metallic Rectangular Bigratings
d
1
x
y
z
λ
gx
λ
gy
d
2
(a) d
2
= 40 nm
d
1
x
y
z
λ
gx
λ
gy
d
2
(b) d
2
= 80 nm
Figure 5.9: Diagram of the effect on the surface profile by increasing the depth
d
2
of
the rectangular bigrating from (a)
d
2
= 40 nm to (b)
d
2
= 80 nm. A larger value of
d
2
increases the diffraction efficiency of the constituent k
gy
grating.
between samples. For the ‘shallow’ bigrating, the target depth of 40 nm was used, for
the ‘deep’ bigrating the target depth was d
2
= 80 nm
The expected effect on the SPP iso-frequency contours is shown in figure 5.10. The
increase in the diffraction efficiency of the
±k
gy
scattering events will be the dominant
effect observed for the chosen grating’s parameters. By increasing the diffractive coupling
between SPPs, the (
±
1
,
0) SPPs will interact and form larger band-gaps at the 1
st
BZ
boundary in the
k
y
direction. This is shown in the figure as a relatively weakly coupled
SPP contour (yellow line) evolving into the stronger coupled SPP contour (red line).
The effect of this is to ‘flatten’ the band along the
k
y
axis. The interactions between
these contours and the (
±
1
,
0) are not included in this picture as they will require a
multiple scattering process by which to interact, and consequently are very weak. A
more detailed description as to the presentation of these bands in
k
-space is discussed
in chapter 4, under the experimental methodology for imaging scatterometry.
Figure 5.11 shows the experimentally mapped iso-frequency contours for the two
rectangular bigratings at a wavelength of
λ
0
= 700 nm. Figure 5.11(a) shows two
SPP contours as a dark bands of reflectivity closely following the
±
1
k
gx
scattered
diffraction cones (blue lines). The SPP contours in figure 5.11(a) exhibit a small degree
of anisotropy with respect to their dispersion, with the SPP contour closer to the
diffracted light lines at
k
y
= 0 than elsewhere along the contour. This is due to the
(
±
1
,
0) scattered SPPs interacting and forming band gaps with the (
±
1
, ±
1) or (
1
, ±
1)
SPPs. This interaction is strong as it only requires a single scattering event of
±
1
k
gy
to
72
5. Optical Response of Metallic Rectangular Bigratings
k
gy
(1, 1) (+1, 1)
(+1, +1)(1, +1)
(+1, 0)(1, 0)
Figure 5.10: A sketch showing the expected iso-frequency contour deformation. Increased
coupling efficiency in to the
k
gy
direction deforms the SPP iso-frequency contours;
represented as increasing in coupling efficiency from poorly coupled together (yellow
lines) to strongly coupled (red). The blue circles represent the diffracted light circles,
and the black circle represents the zero-order (un-scattered) light circle, which is the
area mapped by imaging scatterometry.
(a) d
2
= 40 nm (b) d
2
= 80 nm
Figure 5.11: Experimental iso-frequency contours for two rectangular bigratings at a
wavelength of
λ
0
= 700 nm with (a) shallow orthogonal grooves of
d
2
= 30 nm and (b)
deep orthogonal grooves of
d
2
= 80 nm. The blue lines show the calculated position of
the diffracted light circles, which unperturbed SPP contours will follow.
73
5. Optical Response of Metallic Rectangular Bigratings
couple the SPPs together, which is a harmonic that is present in the grating’s surface
profile. The strength of this interaction causes the deformation in the value for
k
SP P
for different azimuthal angles. This interaction corresponds to the yellow SPP example
contour drawn in figure 5.10, as the (
±
1
,
0) modes which weakly interact with the
(±1, ±1) and (1, ±1) scattered modes.
By deepening the short-pitch grating to 80 nm this anisotropy is increased. This is
seen clearly in the experimental results in figure 5.11(b) where the (
±
1
,
0) scattered
SPPs interaction with the (
±
1
, ±
1) and (
1
, ±
1) scattered SPPs have served to flatten
the SPP contours, pulling them away from their associated diffraction circles.
In both cases the interaction between the (
±
1
,
0) and the (0
, ±
1) scattered SPPs is
not observed as the weak multiple scattering processes do not strongly couple the modes
together. Further, the (0
, ±
1) scattered SPPs are not seen in these scattergrams, as the
polarisation has been selected to highlight only the (
±
1
,
0) scattered modes for clarity.
Of interest is the effect of this increasing band gap on the SPPs travelling solely
along the
k
x
direction. Since the deepening of the short-pitch grating affects only the
diffraction efficiency in the
y
-direction, it is not expected that the SPP travelling solely
in the
x
-direction should be affected. Experiments show, however, that this is not the
case. For the scattergrams in figure 5.11, along
k
y
= 0 we see that the SPP contour
lies further from the diffraction circles in the deep grating case when compared to the
shallow grating. Mapping the dispersion for these two gratings using the zero-order
reflectivity maps as a function of (
ω, k
x
) we obtain the dispersion of the SPP modes in
the plane of incidence containing k
x
, and these are shown in figure 5.12.
The SPP bands in the deep grating case are suppressed in frequency compared to
those obtained from the shallow grating. This frequency shift is seen more clearly in the
spectral plots and overlay of the dispersions shown in figures 5.12(c) and 5.12(d). For
the overlays of dispersion the reflectivity minima were taken as the SPP mode positions
(as it will be travelling along an axis of mirror symmetry, polarisation conversion does
not occur). These were extracted from the spectra of each plot for a given incidence
angle and collated together.
The results show that the effective mode index of the SPP can be altered both in the
k
x
and
k
y
direction by increasing the depth of the short-pitch grating. This is analogous
to the work of Pendry [
72
] and the subsequent work on ‘spoof surface plasmons [
73
],
where surface structure is used to manipulate the asymptotic limit of surface wave
dispersion, introducing the concept of an ‘effective surface plasma frequency’. We shall
return to this conclusion in chapter 7, where we demonstrate that the short-pitch can
be extremely sub-wavelength (
λ
gy
= 150 nm) and still affect the propagation of SPPs
along grating surfaces.
74
5. Optical Response of Metallic Rectangular Bigratings
in-plane wavevector, k
x
(m
1
)
angular frequency, ω (rad s
1
)
0.0
0.2
0.4
0.6
0.8
1.0
1 × 10
6
2 × 10
6
3 × 10
6
4 × 10
6
5 × 10
6
2.5 × 10
15
3 × 10
15
3.5 × 10
15
4 × 10
15
+1
-1
(a) d
2
= 40 nm
in-plane wavevector, k
x
(m
1
)
angular frequency, ω (rad s
1
)
0.0
0.2
0.4
0.6
0.8
1.0
1 × 10
6
2 × 10
6
3 × 10
6
4 × 10
6
5 × 10
6
2.5 × 10
15
3 × 10
15
3.5 × 10
15
4 × 10
15
+1
-1
(b) d
2
= 80 nm
0.0 0.2 0.4 0.6 0.8 1.0
angular frequency, ω (rad s
1
)
R
2.4 2.6 2.8 3 3.2 3.4 3.6 3.8
× 10
15
(c)
in-plane wavevector, k
x
(m
1
)
angular frequency, ω (rad s
1
)
1 2 3 4 5
× 10
6
2.5 3 3.5 4
× 10
15
40 nm
80 nm
+1
-1
(d)
Figure 5.12: (a-b) Experimentally obtained dispersion diagrams mapped from the
reflectance of rectangular bi-gratings with nominal depths of (a) 40 nm, (b) 80 nm. (c)
The SPP mode position measured by reflection of TM polarised light at
θ
= 7
as a
function of angular frequency,
ω
. The curves are for the shallow (black) and deep (red)
grating. The SPP mode shifts to lower frequencies as shown in (d).
75
5. Optical Response of Metallic Rectangular Bigratings
5.6 Conclusions
In this chapter we have introduced the concept of a bigrating as a grating with two
grating vectors which are not collinear. The example used in the chapter was the
rectangular bigrating, which possesses two grating vectors of different magnitude,
oriented at α = 90
.
The dispersion of these rectangular gratings was mapped from the reflectivity of
the samples in section 5.3.2. It was found that small errors in mark-to-space ratio
of such gratings lead to the strong coupling of even-ordered modes, which would not
be expected to couple strongly on such a surface if the mark-to-space ratio was 1. A
discussion as to the features associated with SPPs on such a surface has been presented.
This will be a background to future discussions on dispersion mapping throughout this
thesis.
Band gaps forming between the +1
k
gx
and
1
k
gx
scattered SPPs were experimen-
tally observed using imaging scatterometry, as presented in section 5.3.3, demonstrating
this technique for the first time. FEM modelling of the gratings show the field orientation
for these SPP standing waves which occur at the BZ boundary.
In section 5.4 it was found that such rectangular bigratings exhibit polarisation
conversion at
φ
= 45
, which is different from conventional square bigratings. This is
attributed to the fact that SPPs propagating out of the plane of incidence inhabit an
axis of broken mirror symmetry, and so may decay into either polarisation state.
Finally section 5.5 investigates the effect on SPP dispersion by deepening the short-
pitch of the rectangular bigrating. It is found that the anisotropy of the SPP mode can
be controlled by changing this depth, which controls the strength of coupling between
counter-propagating SPP modes. Finally, the mode index of the in-plane SPP (travelling
along the
k
x
direction) is also found to be affected slightly by the deepening of the
orthogonal pitch.
76
Chapter 6
Optical Response of Metallic
Oblique Bigratings
6.1 Introduction
In this chapter, we examine the propagation of SPPs on periodic surfaces with the
lowest possible order of symmetry. The underlying Bravais lattice for these grating is
oblique, containing no reflection or rotational symmetry operations, save for the one
inversion operation shared by every possible Bravais lattice.
Section 6.2 discusses the oblique grating samples, their fabrication, the coordinate
system used and the reciprocal space lattice for an oblique grating. A comparison
between experiment and a theoretical model for a typical SPP iso-frequency contour
in
k
-space on such a grating is presented in section 6.3. The mapped dispersion of
SPPs on this geometry are recorded, with which their band-structure on the fabricated
oblique gratings are explained. Polarisation conversion mediated by SPPs is also found
on the oblique grating, with SPPs travelling in directions which are not in the plane of
incidence rotating the reflected light’s polarisation.
The most striking result in this chapter is that this low-symmetry class of grating
is shown to be the only exception for the formation of band-gaps at Brillouin Zone
(BZ) boundaries of all the 2D Bravais lattices. We show experimentally in section 6.5
that an SPP propagating across a BZ boundary does not split to form a band-gap, and
offer a generalised symmetry discussion as to why this is so. The dispersion of SPPs
along axes containing the unique points of high-symmetry on an oblique grating are
obtained experimentally, and it is shown that for these discrete planes of incidence, SPP
band-gaps at BZ boundaries are still formed due to the symmetry of these axes.
77
6. Optical Response of Metallic Oblique Bigratings
φ
θ
E
T M
E
T E
d
2
x
y
z
λ
gx
λ
gv
plane of incidence
d
1
Figure 6.1: Coordinate system for an oblique bigrating. The typical design paramters
used in this chapter are λ
gx
= 600 nm, λ
gy
= 400 nm, α = 75
, d
1
= d
2
= 40 nm.
6.2 The Oblique Grating
An oblique lattice is formed of an infinite array of lattice points separated by two lattice
vectors of different magnitudes, oriented at an angle with respect to each other
α
, such
that
α 6
= 90
. To realise this symmetry using surface-relief gratings, two diffraction
gratings of different pitches are ‘crossed’ at an angle
α
such that
α 6
= 90
, forming an
oblique bigrating.
The coordinate system for this type of grating is shown in figure 6.1. The plane of
incidence is defined at an azimuthal angle of
φ
so that when
φ
= 0
the wavevector of
incidence light lies along the
x
-direction. When the electric field vector of the impinging
radiation is contained within the plane of incidence, the light is said to be TM polarised,
and when the electric vector lies orthogonal to the plane, it is TE polarised. The
x
-direction is collinear with the grating vector
k
gx
= 2
π
ˆ
x
gx
for the longer-pitch
monograting, which possesses a periodicity of
λ
gx
. This period, for all the gratings
presented in this chapter, is
λ
gx
= 600 nm. The second, shorter-pitch grating lies at
an angle of
α
= 75
along the
v
axis (defined in the plane of the
xy
plane at an angle
α
to
ˆ
x
) and for the grating presented in this chapter has a period
λ
gv
= 400 nm. As
before, the grating vector of this short-pitch grating is defined as
k
gv
= 2
π
ˆ
v
gv
. An
angle of
α
= 75
was chosen to lie midway between the high symmetry cases of
α
= 60
78
6. Optical Response of Metallic Oblique Bigratings
k
gv
k
gx
Figure 6.2: Example reciprocal lattices for an oblique grating.
λ
gx
= 600 nm,
λ
gv
= 400
nm and the angles between the two periodicities is α = 75
(hexagonal-like) and
α
= 90
(square/rectangular). The fill fraction of the gratings is
designed as Γ
x
= Γ
v
= 0
.
5. The depths of the gratings are
d
1
and
d
2
, and are chosen in
this chapter so that d
1
= d
2
40 nm.
The reciprocal space map of the corresponding lattice is shown in figure 6.2 and
constitutes the lowest symmetry lattice set of all the two-dimensional Bravais lattices.
The reciprocal lattice of the oblique grating is itself oblique, with the reciprocal lattice
vectors defined as
k
gx
and
k
gv
, oriented at an angle
α
?
= 180
75
= 105
with respect
to each other. The only symmetry operation possible for this oblique lattice is a rotation
around a lattice point of 180
, which for 2D lattices is the equivalent of a inversion
operation. Centred about each lattice point are circles representing various scattered
iso-frequency contours. The circles formed with solid lines show the contours for a
grazing photon, and the dashed lines represent the iso-frequency contours for scattered
SPPs which lie outside their respective grazing-photon lines (diffraction lines) due to
the greater momentum of SPPs compared to light. In this simple cartoon, the SPPs do
not interact and cross through each other unperturbed. However, if the SPPs interact
to form band-gaps these iso-frequency contours will deform as detailed in chapter 4.
The black circle is the case of the un-scattered zero-order light, which is the region of
k
-space accessible for mapping using the experimental method of imaging scatterometry.
The gratings for this chapter were fabricated using electron beam lithography (EBL)
79
6. Optical Response of Metallic Oblique Bigratings
Figure 6.3: SEM of an oblique grating master fabricated in a silicon wafer.
and the template stripping method outlined in chapter 4. A scanning electron mircograph
of a fabricated oblique grating master in silicon is shown in figure 6.3. The target param-
eters of this grating were
λ
gx
= 600 nm
,
λ
gv
= 400 nm, Γ
x
= Γ
v
= 0.5, d
1
= d
2
= 40 nm
.
6.3 Coupling of Light and SPP Mode Interaction on Oblique
Gratings
A square profile grating such as those fabricated using EBL contain little or no even-
Fourier components in their surface profile (see chapter 5). Consequently, the momentum
of a plane wave incident on such a grating may be modified with direct scattering events
with values of
±mk
g
, where
m
is odd. A direct scattering event, where the plane wave
is modified by
±nk
g
, where
n
is even, is not possible on a grating where Γ = 0
.
5. There
does exist, however, the opportunity for the plane wave to be modified by a total value
of
±nk
g
through multiple-scattering processes, summing the contributions from
±mk
g
scattering events. These multiple scattering events are found in experiment to be so
weak that they are rarely experimentally observed.
These scattering efficiencies also dictate the strength of interaction between counter-
propagating SPP modes. Modes that are separated by a direct scattering event will
interact strongly. Little, if any, interaction is observed for modes separated by the
weaker multiple-scattering process.
Figure 6.4 shows the theoretical and experimentally measured iso-frequency contours
for an oblique bigrating at
λ
0
= 700 nm. Figure 6.4(a) shows the numerical prediction
using the Chandezon method, approximating the square groove profiles with the Fourier
80
6. Optical Response of Metallic Oblique Bigratings
(a) Model (b) Experiment
Figure 6.4: (a) The theoretically modelled iso-frequency surface for an oblique bigrating
illuminated with TM polarisation at
λ
0
= 700 nm. Mode intersections labelled A-C
are discussed in section 4 (b) Experimentally obtained scattergram of the iso-frequency
contour imaged through crossed polarisers. The yellow circles overlay indicates the
theoretical mode position obtained from (a). Both colour bars range from low reflectivity
(black) to high.
sums,
F (x) =
3
X
n=1
4A
cos
2x
λ
gx
(6.1)
F (y) =
3
X
n=1
4A
cos
2v
λ
gv
(6.2)
Providing a suitable approximation to a lamella bi-grating with a depth of 2
A
= 40 nm,
λ
gx
= 600 nm,
λ
gv
= 400 nm,
α
= 75
and Γ = 0
.
5. The dielectric function of silver
is taken from literature [
84
], and for the illuminating wavelength of 700 nm is equal
to
ε
=
23
.
13 + 0
.
59
i
. The light in the theoretical plot is TM polarised for every
azimuthal angle. Only the
n
= 1
,
3 components are included in the calculation, as the
even components are considered to be absent for a grating with Γ = 0.5.
The corresponding experimentally obtained iso-frequency surface is shown in figure
6.4(b). The contour is obtained using imaging scatterometry detailed in chapter 4. For
the illuminating wavelength of 700 nm, the contrast of the entire SPP contour to the
background is weak, making the determination of the mode position in
k
-space difficult.
81
6. Optical Response of Metallic Oblique Bigratings
To improve the contrast, the polarisers of the scatterometer are crossed, producing a
dark background reflectivity against which the polarisation conversion mediated by the
SPPs [
34
] provides greater contrast for the mode positions. The four bright lobes of
high reflectivity is polarisation conversion mediated by the ellipsoidal mirror in the
apparatus (see chapter 4).
The modelled values for the mode position (taken from the theoretical plot at the
position of reflectivity minima) found in figure 6.4(a) are plotted on the experimental
results in figure 6.4(b) as yellow circles, and show good agreement, especially considering
the simplicity of the model. This shows that the dominant scattering amplitudes in the
observable SPP band structure are only the n = 1, 3 components.
In the upper half-space of figure 6.4(a), three SPP contour crossings are labelled
A-C
. Crossing point A is the meeting of a (
1
,
0) and a (0
,
1) Bragg scattered SPP
contour. These two SPP curves are separated in
k
-space by a minimum of two scattering
vectors, and so require a multiple scattering process to interact with each other. With
no 2
nd
order harmonics in the grating profile, the interaction is weak and no band-gap
is observed. The crossing point B is the intersection of the (0
,
1) and (1
,
1) SPPs. This
is a process by which the SPPs must scatter a total of 1
k
gx
to interact, and so a small
band-gap forms. At point C, the (1
,
0) and (1
,
1) SPP cross. These are separated by a
single scattering vector 1k
gv
, and so interact, forming a large band-gap.
The coupling efficiencies of plane-polarised light to all these SPP modes is strong
as the majority of the contours present are Bragg scattered back into the light circle
by a single scattering vector, which is inherent in the grating profile. The exception
is the (1
,
1) SPP, (the contour between points B and C), which is scattered back in to
the light circle by a multiple scattering process. It is, however, observed quite strongly
between points B and C, though not strongly (and observed as a weak bright band)
outside of these points.
This coupling between points B and C is an example of SPPs ‘self-coupling’, the (0
,
1)
and (1
,
0) SPPs coupling to the SPP contour close to the crossing points where there
is a component of the SPPs that fulfil the required energy and momentum matching
conditions for the (1
,
1) mode. These well coupled (0
,
1) and (1
,
0) SPPs resonantly
drive the (1, 1) SPP, resulting in its observation as a dark band.
These interactions can also be observed by mapping the dispersion relation of
the SPPs by recording the zero-order reflectivity from the sample as a function of
polar angle and wavelength. TE polarised light is incident on the sample and the
reflectivity measured for the wavelength range
400 nm < λ
0
< 850 nm
and the angle
range
7
< θ < 75
(see chapter 4). These experimental results are shown in figure
These mode crossings are equivalent through an inversion symmetry operation to the crossings in
the lower half-space
82
6. Optical Response of Metallic Oblique Bigratings
Figure 6.5: The dispersion of modes on an oblique grating for
φ
=
α
= 75
, measured
using the reflectivity as a function of polar angle and wavelength. The mode posi-
tions (grayscale minima) and calculated diffraction edges (lines) have been coloured to
correspond to the scattered mode labels given in table 6.1.
(n, m) (-1,0) (1,0) (1,1) (-1,1) (0,1)
(-1,0) X
(1,0) X
(1,1) X X
(-1,1) X
(0,1) X
Light X X X
?
Table 6.1: The expected strong coupling (
X
) between SPP modes. The colour of
the (
n, m
) mode labels correspond to the highlighted modes in figure 6.5 scattered by
(
nk
gx
, mk
gv
).
X
?
indicates coupling, but only for the orthogonal polarisation (TM) and
so is not observed in figure 6.5.
83
6. Optical Response of Metallic Oblique Bigratings
6.5 for an azimuthal angle of
φ
= 75
, with the plane of incidence containing
k
gv
.
To facilitate the discussion of mode interactions, the dark-bands of low reflectivity
which map the SPP dispersion have been colour coded depending upon which dominant
scattering event the SPP contour is associated with. The associated diffraction lines
have also been coloured accordingly. These scattering interactions between the present
modes are summarised in the companion table 6.1.
The band gaps are clearly observed for the interactions between the (1
,
0) scattered
SPP band (green) and the (1
,
1) scattered SPP band (blue). Another large band-gap
is observed for the interaction of the (
1
,
1) scattered SPP (cyan) and the (
1
,
0)
scattered SPP (red), which again are SPPs separated in
k
-space by a single grating
vector of 1k
gv
.
A smaller band-gap is also observed for the (
1
,
0) and (1
,
1) SPP crossing (red and
blue bands, respectively). This coupling is mediated either by multiple scattering events
or a weak direct-scatter due to a small deviation in Γ away from the designed value
of 0.5, and consequently the frequency gap formed is small in comparison to the other
observed band-gaps.
The presence of a dark band associated with the (1
,
1) scattered SPP is clearly
observed in the mapped dispersion and coloured as a blue band of low reflectivity,
despite the (1
,
1) scattering event requiring multiple scattering with which to interact
with zero-order light. This region of the dispersion is observed for the same reason the
iso-frequency contour between points B and C in figure 6.4 is observed, a component
of the (
1
,
0) and (1
,
0) SPP fields present on the surface resonantly drive this mode.
The SPPs have ‘self-coupled’. A very small reflectivity minimum (not coloured) lies
between the red (
1
,
0) and the green (1
,
0) scattered SPP bands. There is also weak
self-coupling of these modes to the in-plane (0
,
1) SPP (its presence cannot be attributed
to the incident light field, as the incident light’s polarisation state is incorrect for its
excitation for this orientation).
6.4 Polarisation Conversion
In this section we discuss how SPPs on an oblique bigrating mediate the rotation of the
incident field’s polarisation. An example is experimentally demonstrated in figure 6.6.
This figure shows the reflectivity of an oblique grating as a function of (
ω, k
v
),
mapping the SPP dispersion by the characteristic minima in reflected intensity. For
subfigure 6.6(a), the incident polarisation is set to be TE polarised, and the detector’s
polariser is also set to TE. The corresponding polarisation conversion signal is recorded
by rotating the detector’s polariser by 90
to TM, and is shown in figure 6.6(b). The
bright bands of polarisation converted signal lie along the same SPP contours, showing
84
6. Optical Response of Metallic Oblique Bigratings
in-plane wavevector, k
v
(m
1
)
angular frequency, ω (rad s
1
)
0.0
0.2
0.4
0.6
0.8
1.0
0.2 0.4 0.6 0.8 1 1.2
× 10
7
2.5 3 3.5 4
× 10
15
(a) R
T E:T E
, φ = 75
in-plane wavevector, k
v
(m
1
)
angular frequency, ω (rad s
1
)
0.00
0.01
0.02
0.03
0.04
0.05
0.2 0.4 0.6 0.8 1 1.2
× 10
7
2.5 3 3.5 4
× 10
15
(b) R
T E:T M
, φ = 75
Figure 6.6: (a) Polarisation conservation and (b) conversion mediated by out-of-plane
scattered SPPs on an oblique grating.
that it is the surface waves which mediate this polarisation conversion.
The polarisation converted signal is normalised to the reference spectrum with both
polarisers set to TE polarised, and with this definition, only a maximum of
5% of
the light coupled into the SPPs is re-radiated in to the orthogonal polarisation state.
Additionally, the SPP modes in the plane (which travel collinear to the wavevector
k
gv
),
do not participate in this polarisation conversion as the SPPs are travelling along a
mirror-plane of the grating associated with the
k
gv
direction (although there are no
mirror planes associated with the overall bigrating structure). This is an important point:
SPP mediated polarisation conversion occurs only when the SPPs are travelling along the
surface in a direction out of the plane of incidence and along a direction of broken mirror
symmetry. An oblique bigrating always provides out-of-plane scattering mechanisms
and so polarisation conversion will always be observed for out-of-plane scattered SPPs.
If, however, the plane of incidence contains a grating vector, conservation of momentum
dictates that the SPP associated with this grating vector will travel along in the plane of
incidence and polarisation conversion does not occur, even though the SPP is traversing
a surface of broken symmetry.
6.5 SPPs at the BZ Boundary of Oblique Bigratings
The formation of photonic band-gaps on an oblique bigrating is of interest, as in this
section we present results that show that on a surface with such broken symmetry
85
6. Optical Response of Metallic Oblique Bigratings
the locations in
k
-space of SPP standing waves do not necessarily occur at the BZ
boundary. This concept is known for the band-gaps that occur for electron propagation
in semiconductor crystals [
120
], but has not previously been demonstrated for SPPs on
gratings. To show this clearly, we must identify how photonic band-gaps are illustrated
in the iso-frequency contours recorded with imaging scatterometry. The iso-frequency
image obtained using scatterometry maps
k
-space at a single frequency, with SPP bands
seen as an anomaly in the reflected light. The group velocity of a general propagating
wave is defined as,
v
g
=
k
ω(k) (6.3)
where
v
g
is the group velocity,
ω
(
k
) is the angular frequency of the wave as a function
of wavevector,
k
, and
k
is the gradient operator with respect to
k
. For a small change
in frequency
, the corresponding small movement in
k
-space,
dk
, is related to this
group velocity simply by,
=
k
ω(k) ·dk (6.4)
For an iso-frequency contour, there must be no change in frequency along the contour
(
= 0). Setting
= 0 restricts the values of
dk
to those values that move along a
contour of equal frequency. It is then apparent that in the resultant expression,
v
g
·dk = 0 (6.5)
that
v
g
lies perpendicular to
dk
. This is true for any general contour of constant
frequency. If the group velocity in one direction falls to zero at a boundary, such
as at the BZ boundary, the iso-frequency SPP contour will intersect that boundary
perpendicularly.
Plasmonic band-gaps on surface relief gratings are covered in chapter 2. To briefly
recap, when a SPP meets an equivalent, counter-propagating SPP a standing wave forms.
There are generally two possible arrangements of the electric field for a SPP standing
wave on a grating, which will generally differ in energy. This leads to an upper and
lower energy band, with an energy range between where SPP propagation is forbidden.
The energy gap size is dependent on the energy of the two possible field distributions,
and so is linked intimately to the surface geometry. The surface profile also provides the
scattering mechanism by which SPPs Bragg scatter to meet counter-propagating SPPs
and form these standing waves. The strength of this scattering, and so the amplitude of
the Bragg scattered SPP, was discussed earlier in chapter 5 and affects the magnitude
in energy of the band-gap.
Figure 6.7 shows the mapped iso-frequency contours of SPPs for the oblique grating
at a wavelength of
λ
0
= 650 nm. The position of the SPP contours are found to present
86
6. Optical Response of Metallic Oblique Bigratings
Figure 6.7: An experimentally obtained iso-frequency contour map of SPPs on an oblique
bigrating with illuminated with incident light of wavelength
λ
0
= 650 nm. SPP positions
are mapped as regions of low reflectivity. The blue circles indicate diffracted light circles,
and the green region is the typical BZ boundary drawn using the Wigner-Seitz method.
(a) (b)
Figure 6.8: Regions of interest from figure 6.7. (a) The (
1
,
0) scattered SPP contour
at the BZ boundary and (b) the (+1, 0) scattered SPP contour at the BZ boundary.
87
6. Optical Response of Metallic Oblique Bigratings
as bands of low reflectivity, with the polariser of the experiment chosen in this case to
best couple light to the (
±
1
,
0) modes. Also annotated on the figure are the diffracted
light circles (blue lines) and the BZ boundary formed using the Wigner-Seitz method
[75] (green line).
It is observed that the SPP contour passes through the BZ boundary seemingly
unperturbed. Figure 6.8 shows the two regions in which this occurs in greater detail.
In figure 6.8(a) the SPP contour following the (
1
,
0) diffracted light circle is shown
to pass through the BZ at an angle which is not perpendicular to the boundary. This
means that at the boundary, the SPP’s group velocity in the (0
,
1) direction is not zero,
and no standing-wave states in this direction have formed. The (0
,
1) scattered SPP is
not observed in this figure as the polarisation of the illuminating light has been chosen
to only couple strongly to the (1
,
0) SPP. Additionally the (
1
,
0) and (uncoupled)
(0
,
1) SPPs are not seen to interact, separated as they are by a weak multiple scattering
process.
The same behaviour is observed in the second region in figure 6.8(b). Here, a
band-gap has formed inside the BZ
, but the group velocity at the BZ boundary is still
finite. This highlights the fact that the mode crossing positions on an oblique grating
do not coincide with BZ boundary constructed using the Wigner-Seitz method.
A Brillioun Zone boundary in reciprocal space outlines a unit cell in the reciprocal
lattice and contains on the boundary points of high-symmetry. One way to determine a
boundary that contains the maximum amount of high-symmetry points is to connect
the perpendicular bisectors of the vectors connecting the nearest neighbours to one
lattice point, a method known as the Wigner-Seitz method. The area mapped out by
this method, for the highly-symmetric cases of square, rectangular or hexagonal lattices,
is exactly equivalent to the area we call the Brillioun Zone, and no other choice of unit
cell contains as many points of high-symmetry in the boundary.
In the oblique case, we may again use the Wigner-Seitz method to construct a
perfectly valid BZ, shown in figure 6.9 as the area bounded by a red line. In this figure,
it is seen that the only points of high symmetry on the oblique lattice are the singular
points lying midway along the perpendicular bisectors, and are labelled
i
1
, i
2
and
i
3
.
These points are equivalent to the points
i
1
, i
2
and
i
2
by a rotation operation of 180
,
and are isolated points of symmetry. However, this unit cell is not a unique choice, we
can easily contain these points of high-symmetry lying on the boundary of a unit cell
by using instead a trapezoidal unit cell, as shown by a blue line. In conclusion: The
most primitive unit cell of an oblique lattice is not unique.
The arbitrary choice of the unit cell which contains the high symmetry points, and
Recall that the SPP eigenmodes are unaffected by the coupling of the light, and so although some
branches of the SPP contours are uncoupled to light in this scattergram, the band-gap still exists.
88
6. Optical Response of Metallic Oblique Bigratings
Figure 6.9: Two possible primitive unit cells for an oblique lattice. The Brillioun Zone
(Wigner-Seitz) cell (red) and a simple trapezium (blue) contain on their boundaries the
points of high symmetry labelled i
1
, i
2
, i
3
.
that both choices is as useful as the conventional BZ in the oblique case, highlights
that the properties of physical phenomena on the lattice are determined, not by the BZ
boundary, but by the symmetry operations of the lattice.
Neumanns principle with respect to our system requires that the physical properties
of phenomena associated with the grating possess the same symmetry as the point
symmetry group of the grating [
121
]. While we could discuss specific propagation
properties of the surface modes with respect to its lattice, let us instead generalize
these concepts to some arbitrary vector field and see what restrictions the symmetry
of the grating places upon it. Figure 6.10 shows an arbitrary vector,
r
lying on the
boundary of a rectangular or oblique unit cell. In the case of a rectangular grating,
the mirror and translational symmetry allows the deduction of the other shown vectors
through various reflections in the
σ
v
and
σ
h
planes or rotations about the
C
point.
These vectors sum to give a magnitude of zero in the direction perpendicular to the zone
boundary. Whether this vector field represents the SPPs momentum, group velocity
or Poynting vector, the conclusion is the same: a standing wave forms perpendicular
to the BZ boundary. In the plasmonic case, a SPP has Bragg-scattered and interfered
with a counter-propagating SPP. The interference between these two SPPs, which by
symmetry can be equal in magnitude and opposite in direction, forms a standing wave.
The symmetry considerations we have outlined show us that this is possible, with no
restrictions on a mirrored SPP occupying the same position and, importantly, that
their vector properties will have equal magnitudes and so interfere completely. The two
possible field arrangements for SPP standing waves on gratings lead to two solutions of
different energies with a forbidden band of no propagating waves between the two: a
89
6. Optical Response of Metallic Oblique Bigratings
Figure 6.10: Applying the symmetry operations of the rectangular BZ (left), an arbitrary
vector,
r
(lying on the BZ boundary) corresponds to seven other vectors of known
magnitude and direction. The summation of these vectors leads to no perpendicular
component of the vector at the BZ boundary. In the oblique case (right), there is no
such condition, the
C
2
rotation operation only placing constraints on three additional
vectors.
plasmonic band gap. These are observed experimentally as discontinuities of the SPP
curves at the BZ boundary, as was seen in chaptrer 5.
Using the same approach, we apply the symmetry operations of the oblique lattice
to an arbitrary vector field in figure 6.10. With no mirror symmetry, the oblique lattice
possesses only translational and a two-fold rotation symmetry operations (a two fold
rotation operation is equivalent in two dimensions to an inversion operation). As shown
in the figure, there are no special conditions on the vectors lying along the BZ boundary
formed using the Weigner-Seitz method, and no condition for the vectors to cancel
perfectly. Standing waves do not necessarily occur at the BZ boundary.
This leads to the observation that the band-gaps observed do not form at the BZ
boundaries, but simply where the SPP meets a Bragg scattered counter-propagating
SPP. The mid-points of BZ boundaries are of high symmetry in both rectangular and
oblique cases (points
i
1
, i
2
, i
3
in figure 6.9), and do allow the total cancellation of our
vector field. We can observe the points of zero group velocity at these BZ boundaries
by mapping the dispersion of the SPP mode in the special cases of
φ
= 0
and
φ
= 75
,
which place the plane of incidence intersecting points
i
1
and
i
2
in figure 6.9. In these
cases, we fully expect a band-gap to be present at the BZ boundary. These plots are
shown in figure 6.11. In the case of of a dispersion diagram such as this, zero group
velocity is shown as ω/∂k
k
= 0.
90
6. Optical Response of Metallic Oblique Bigratings
in-plane wavevector, k
x
(m
1
)
angular frequency, ω (rad s
1
)
0.0
0.2
0.4
0.6
0.8
1.0
0.2 0.4 0.6 0.8 1 1.2
× 10
7
2.5 3 3.5 4
× 10
15
π λ
gx
(a) R
T M
at φ = 0
in-plane wavevector, k
v
(m
1
)
angular frequency, ω (rad s
1
)
0.0
0.2
0.4
0.6
0.8
1.0
0.2 0.4 0.6 0.8 1 1.2
× 10
7
2.5 3 3.5 4
× 10
15
π λ
gv
(b) R
T E
at φ = 75
Figure 6.11: Dispersion plots mapped using the reflectivity of an oblique grating, showing
the occurrence of band-gaps at the Brillioun Zones along the special cases of (a)
φ
= 0
and (b) φ = 75
.
6.6 Conclusions
In this chapter, SPPs propagating on an oblique bigrating have been investigated. The
dispersion of these surface modes has been mapped and the SPP interactions discussed
in terms of the available scattering amplitudes of the grating. Polarisation conversion
is observed on this grating, as SPPs travelling in a direction other than the plane of
incidence mediate the rotation of the light’s polarisation state.
Using imaging scatterometry, it is observed that the SPP contours are not perturbed
as they pass through the conventional BZ boundary. A generalized discussion on the
symmetry of the BZ is presented, concluding that this is because the BZ boundary on
an oblique grating is not a contour of high symmetry, and only contain isolated points
around which the symmetry conditions may be met for the formation of SPP standing
waves. Finally, when the plane of incidence intersects these unique points, SPP band
gaps may still be observed.
91
Chapter 7
Optical Response of Metallic
Zigzag Bigratings
7.1 Background
This chapter introduces a novel type of diffraction grating, the zigzag grating. A detailed
experimental and theoretical study of the SPPs excited on such a grating is explored,
with interesting results pertaining to the polarisation requirements of the coupled light,
the plasmonic band-gap character of interacting SPP modes, and the highly anisotropic
propagation of SPP on such a grating.
In 2005, Kleemann et al. [
122
] presented a new method for designing diffractive
optical elements with efficiencies and properties comparable to traditional surface relief
gratings. These gratings use sub-wavelength features perpendicular to the diffractive
periodicity to control the Fourier components available to scattering light [
123
]. The
symmetry and structure of these sub-wavelength features control the diffraction efficien-
cies on such a grating, the way traditional surface relief gratings may ‘blaze’ their groove
profile to achieve the same result [
32
]. These diffraction gratings can be produced using
well established lithography techniques to pattern the surface, a simpler task than pro-
ducing complicated and precise groove shapes on traditional blazed gratings. Kleemann
at al. named these gratings Area Coded Effective medium structures (ACES), and the
versions exhibiting similar characteristics to blazed surface-relief gratings ‘BLACES’.
Surface Plasmon Polaritons were excited on such BLACES in 2009 by Bai et al.[
95
] and
showed that asymmetrical excitation of the SPPs could be achieved at normal incidence,
typical of the grating’s blazed character. It is established by this previous work that a
patterned metallic surface with sub-wavelength features perpendicular to the diffractive
periodicity is capable of manipulating the strength of diffracted orders and, consequently,
the coupling of SPPs to light. The use of the sub-wavelength structure to manipulate
92
7. Optical Response of Metallic Zigzag Bigratings
the optical response of these surfaces qualifies them as a type of ‘metamaterial’ surface
[124].
In the work to date on BLACES, the propagation of the SPPs has been along
an axis of mirror symmetry. Other work has hinted at novel optical effects, such as
‘magnetic mirrors’ [
125
,
126
] or spatial control of coherent anti-Stokes emission [
127
],
when plasmonic resonances are coupled to by light on geometries of broken symmetry.
In this chapter, we present a new ACES grating with sub-wavelength structure with
a diffracting periodicity along an axis of broken mirror-symmetry. We name this grating
a ‘zigzag’ grating. Coupling of light to SPPs on this grating is highly polarisation
selective and both TM and TE polarised light can excite SPPs propagating in the same
plane, which is theoretically investigated in section 7.3.1 and experimentally verified
in sections 7.3.3 and 7.3.4. Further, we find that the symmetry of this grating places
restraints on the formation of plasmonic band-gaps at Brillouin Zone (BZ) boundaries,
which is explored in section 7.4 and that the propagation of SPPs is highly anisotropic
with respect to azimuthal angle (section 7.5). The degree of SPP anisotropy is so great,
that the zigzag grating is suggested as an excellent candidate for single-wavelength
collimation of SPP beams for use in plasmonic circuits. This highly anisotropic SPP
band structure leads to the observation of out-of-plane SPPs associated with the sub-
wavelength periodicity, which evolve to have cavity-resonance character for deep zigzag
gratings.
7.2 The Zigzag Grating
A diagram of a zigzag grating is shown in figure 7.1, including the coordinate system
used throughout this chapter. The zigzag grating is formed of a surface-relief grating
of sub-wavelength, and hence non-diffracting grooves that run along a silver surface.
Perpendicular to this non-diffracting grating, the grooves are perturbed in the surface
plane to introduce a long-pitch variation that may diffract visible light. This zigzag
perturbation introduces a diffracting pitch of
λ
gx
, which lies perpendicular to the short-
pitch of the surface-relief grating,
λ
gy
. In many ways, the zigzag geometry is similar
to a conventional rectangular bi-grating, however, the long-pitch is present due to a
zigzag surface perturbation (not surface-relief grooves). This makes the zigzag grating’s
optical response quite unlike that of a standard bigrating .
The plane of incidence is defined at an azimuthal angle,
φ
, which equals 0
when the
plane is coincident with the plane of diffraction from the long pitch. The polar angle
θ
of impinging radiation lies in the plane of incidence, and is defined as
θ
= 0
when
the light is incident normal to the average plane of the surface. The polarisation of the
light is defined as Transverse Magnetic (TM) polarised when the electric field vector
93
7. Optical Response of Metallic Zigzag Bigratings
φ
θ
E
T M
E
T E
d
x
y
z
λ
gx
λ
gy
plane of incidence
Figure 7.1: The coordinate system of a zigzag grating. The experimental sample
parameters were
λ
gx
= 600 nm,
λ
gy
= 150 nm,
d
= 29
.
9 nm with the plane of incidence
defined at an angle
φ
and the polar angle of incidence as
θ
. The two polarisation
orientations for the electric field vector are also shown for TM and TE polarisations.
lies in this plane of incidence, and Transverse Electric (TE) polarised when the electric
field vector lies orthogonal to it. The zigzag grating under consideration in this chapter
possesses mirror symmetry along the yz planes defined at x = 0, λ
gx
/2, λ
gx
.
7.3 The Coupling of Plane Polarised Light to SPPs on
Zigzag Gratings
7.3.1 Theory
To examine how the orientation of the electric field vector of impinging radiation may
relate to the excitation of SPPs on a zigzag grating, we consider the magnitude of the
electric vector that lies normal to the zigzag surface for the two polarisation cases of TM
and TE polarised light. The induced local surface charge is, to a good approximation,
proportional to the electric vector normal to the surface [
128
], so by considering how the
local normal of electric field varies along the grating, we may infer what local surface
charge arrangements are possible on such a grating. At normal incidence (
k
x
= 0), the
wavevector of these different charge distributions are then equal to the Bragg vectors
94
7. Optical Response of Metallic Zigzag Bigratings
0
1
2
3
4
5
6
0
1
2
3
4
5
6
(a)
E
ˆn
for
p
=
ˆ
x
polarised light
0
1
2
3
4
5
6
0
1
2
3
4
5
6
(b) E
ˆn
for
ˆ
y polarised light
Figure 7.2: The surface normal components of electric field vector for (a) TM and (b)
TE polarised light (arrows) projected on a contour of the zigzag surface profile. The
contour plot amplitude ranges from 1 (white) to -1 (blue).
required of the zeroth order SPP to scatter and interact with light.
A simple expression for the local normal component of electric field can be obtained
by considering an approximation to the zigzag surface given by,
r = x + y + cos (y cos x) ˆz (7.1)
representing a zigzag profile having unit amplitudes, and periodicities of 2
π
. At normal
incidence to the
xy
plane, the polarisation vector of the electric field is defined as
p
=
ˆx
for TM and p = ˆy for TE polarisations. The normalized surface normal function is
ˆn =
x
r ×
y
r
|
x
r ×
y
r|
(7.2)
To induce surface charge density at the interface, the electric field will require a
component normal to the surface. The electric vector lying normal to the surface is
then simply,
E
ˆn
= (ˆn ·p) ˆn (7.3)
The normal electric field vector for each polarisation case is shown projected on
to the approximated zigzag surface in figure 7.2. It is clear that both TM and TE
polarisation vectors result in a surface normal component of electric field on the zigzag
surface, and so may induce surface charge.
We can now examine the functional form of the allowed normal electric field in the
propagation direction for a SPP. The components,
E
T E
and
E
T M
, of the electric field
95
7. Optical Response of Metallic Zigzag Bigratings
x
E
E
T E
E
T M
0 π 2π
Figure 7.3: The magnitude of the surface normal electric field in the x-direction for TM
and TE polarisations.
normal to the surface, lying along the direction of propagation (the
ˆx
direction) and
integrated over y for the two polarisation cases are found to be,
E
T M
=
4π sin
2
(x)
cos (2x) 3
(7.4)
E
T E
=
4π sin (x)
cos (2x) 3
(7.5)
Both
E
T E
and
E
T M
are non-zero, so we may conclude that either polarisation may
induce surface charge and possibly excite SPPs. Plots of
E
T M
and
E
T E
in figure 7.3
show that the electric field vector normal to the surface varies spatially twice as fast for
the TM case as for the TE case. This leads to the conclusion that the wavevector of a
TM-coupled SPP is required to be twice that of the TE-coupled case. By expanding
both these expressions as a Fourier sum in
x
, we may express the
E
T M
and
E
T E
as a
sum of plane-waves. It is possible then to find which in-plane wavevectors of incident
light are required to match these field profiles. Doing so yields,
E
T E
=
X
n=1,3,5,...
a
n
cos (nx) (7.6)
E
T M
= a
0
+
X
n=2,4,6,...
b
n
sin (nx) (7.7)
96
7. Optical Response of Metallic Zigzag Bigratings
where
a
n
and
b
n
are the Fourier series coefficients. For incident light to match
E
T E
requires a series of only odd-ordered terms, while
E
T M
requires a series of only even-
ordered terms. Diffracted fields at the surface will contain both odd and even wavevector
components. Equations 7.6 and 7.7 predict that TE polarised light will provide a suitable
electric field distribution to enable the excitation of SPPs via only odd-ordered diffracted
orders, while TM polarised light will excite SPPs only via even-ordered diffracted orders.
This concept will be discussed again with reference to the experimental and modelled
results in sections 7.3.3 and 7.3.4.
We can visualise these coupling conditions in a simplistic diagrammatic way, shown
in in figure 7.4. Choosing the electric field vector to be at a single instance in time, we
consider the effect on the charge carriers under the influence of this (effectively DC)
field. The arrangement of these charges is illustrated in figure 7.4. These arrangements
lead to an electric potential across the width of the grooves, with zero field along the
lines connecting the zigzag apexes (red dots). The component of the respective electric
fields along the plane of incidence (red arrows) is clearly different in both cases, with
the TM field distribution varying twice as fast as the TE distribution in the plane of
incidence. Crucially, neither solution for the TE or TM field arrangement along the
x
-axis is zero, meaning that charge density may be induced by either TE and TM
polarisations. Returning to the periodicity of the fields, it is clear that the TM case
has a wavelength equal to
λ
gx
/
2, while the TE case has a wavelength equal to
λ
gx
.
To resonantly drive these fields with incident light, the impinging field must match
the wavevector of these surface plasmon field distributions (2k
gx
for the TM case, k
gx
for the TE case). This is achieved via diffraction coupling, with first order diffraction
k
0
±k
gx
matching the fields of the TE case, and the second order diffraction,
k
0
±
2
k
gx
matching the field of the TM case. As such, the polarisation selectivity of coupling
diffracted light to the surface plasmons is demonstrated.
7.3.2 Samples
A schematic of the experimental sample is shown in figure 7.1, comprising of a grating
with a 600
±
5 nm zigzag pitch (
λ
gx
) to provide diffractive coupling to the SPPs. The
periodicity of the non-diffracting surface-relief grooves (
λ
gy
) is 150
±
5 nm. A master of
the zigzag grating was produced in silicon using electron beam lithography, as detailed
in chapter 4. The pattern was exposed in PMMA using a write field of 400
µ
m and a
write field stitching error on the order of 20 nm. The exposed pattern was developed
and reactive ion etched into a Si master. This produced a 2
.
5 mm
2
grating on a silicon
wafer. A scanning electron micrography of one such master is shown in figure 7.5(a).
Results from two different versions of this zigzag grating sample are presented in this
97
7. Optical Response of Metallic Zigzag Bigratings
x
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
E
TE
+
+
+
+
+
+
+
+
+
+
+
+
TM
E
λ
g
+E
x
0 E
x
λ
g
2
λ
g
+E
x
0 E
x
Figure 7.4: Schematic cartoon of light coupling to SPPs on a zigzag grating. The top
cartoons depict a zigzag cavity bounded by two metal zigzag ‘peaks’ (hashed areas)
for a typical zigzag grating as investigated in this chapter. The left images show the
case of TE polarised light, with the TM case on the right. The field applied for both
polarisation cases lead to positive (+) and negative (
) charge distributions along the
grooves. The component of electric field in the
x
-direction orginating due to these
charge arrangements is shown in red.
98
7. Optical Response of Metallic Zigzag Bigratings
(a) (b) (c)
Figure 7.5: Scanning electron micrographs of: (a) an example template-strip master in
Si, used for production of zigzag gratings; (b) a polymer replica of a Si master prior to
metallisation, for zigzag gratings embedded in glass; (c) a template stripped sample in
silver.
chapter. One is a zigzag grating produced by the template stripping method detailed
in chapter 4, the SPPs propagating along the air/metal interface. The second sample
is the same zigzag grating embedded in glass, so that the SPPs propagate along the
glass/metal interface. The reason for the two samples is that observation of novel
higher-frequency modes, such as the 3
rd
order diffracted SPPs, may be out of the visible
domain when the SPP propagates along the air/metal interface. Using a higher index
bounding material, the phase velocity of the SPPs is reduced and so for the same value
of in-plane momentum, these modes may be brought down in frequency into the visible
domain. The glass/grating sample was produced using a slightly different embossing
method, detailed below. In each section, the grating used is clearly stated at the start.
For the glass/grating sample, the structure was duplicated in UV-cured polymer.
Scanning electron microscopy was used to image the surface of this zigzag grating
replica in the polymer (figure 7.5(b)), prior to metallisation. From this micrograph,
the mark-to-space ratio of the final grating structure was determined to be 0.75 with
λ
gx
= 600 ± 5 nm
, and
λ
gy
= 150 ± 5nm
. This pattern was then transferred to an-
other UV cured polymer (Norland Optical Adhesive 73) adhered to a glass substrate
(
n
632.8 nm
= 1.518
) using an embossing method. Silver was thermally evaporated under
high vacuum (5
×
10
6
mbar) with
50 nm
depositions followed by 10 minutes relaxation
time repeated until an optically thick layer of
200 nm
was recorded on the quartz crystal
thickness monitor. This stepped procedure was used to prevent the temperature rising
on the substrate enough to damage the polymer. The substrate was then adhered to
the back of a prepared glass hemisphere of radius 21.75 mm and refractive index
n
= 1.518 using index matching fluid, the base of the hemisphere having been modified
99
7. Optical Response of Metallic Zigzag Bigratings
in-plane wavevector, k
x
(m
-1
)
angular frequency, ω (rad s
-1
)
0.5
0.6
0.7
0.8
0.9
1.0
-1
+1
+2
0
0 0.2 0.4 0.6 0.8 1 1.2
× 10
7
2.5 3 3.5 4
× 10
15
(a)
in-plane wavevector, k
x
(m
-1
)
angular frequency, ω (rad s
-1
)
0.0
0.2
0.4
0.6
0.8
1.0
-1
+1
+2
0
0 0.2 0.4 0.6 0.8 1 1.2
× 10
7
2.5 3 3.5 4
× 10
15
(b)
Figure 7.6: Experimental data (a) and FEM model prediction (b) of TE polarised
reflectivity as a function of in-plane wavevector and angular frequency, mapping SPP
dispersion on a zigzag grating. Blue lines show the positions of diffracted light lines
scattered by mk
gx
, where m = ±1, +2, 0
by removing
1 mm
to account for the affixed substrate thickness. This arrangement
allows illumination of the embedded grating structure at the metal/glass interface at all
angles of incidence. The mounting also protects the surface of the silver from sulphur
contamination [129].
7.3.3 Transverse Electric Coupling
The dispersion of the SPPs supported on a zigzag grating is mapped as a function of in
plane wavevector,
k
gx
, by varying the polar angle
θ
and angular frequency,
ω
, using a
polarised, collimated, monochromatic beam, produced by a white light source together
with a spectrometer. The wavelength range used was 400
< λ
0
<
850 nm with the angle
range 7
< θ <
60
(see chapter 4). When the energy and in-plane momentum of the
light matches that of a SPP mode, the light couples to a SPP and an extremum of the
reflectivity is observed. The experimental and theoretical reflectivity of incident TE
polarised light at φ = 0
is shown in figure 7.6, for a zigzag grating in air.
Two dark bands of low reflectivity are observed associated with the
±k
gx
light lines
(blue lines). These bands map the position in
ω k
x
space of SPPs scattered by grating
vectors equal to
k
SP P
=
k
x
±k
gx
. As predicted by equation 7.7, this odd-scattered mode
is coupled to by TE polarised light. The dispersion of the mode in the experimental
data (figure 7.6(a)) and theoretical model obtained using FEM (figure 7.6(b)) show
excellent agreement. The model parameters used were for a depth of 70 nm and a
100
7. Optical Response of Metallic Zigzag Bigratings
15 20 25 30 35 40
0.4 0.5 0.6 0.7 0.8 0.9 1.0
polar angle, θ (
)
Reflectivity
R
T E
R
T M
0.8 0.9 1
15 40
Figure 7.7: Specular reflectivity of TE (black) and TM (red) polarised light as a function
of polar angle,
θ
, for a wavelength of 632.8 nm,
φ
= 0
. Circles: recorded data with
error bars of 1%, line: fitted FEM model prediction.
mark-to-space ratio between the sub-wavelength grooves of 1, using the silver dielectric
function as reported by Nash & Sambles [
84
]. The experimental sample is expected
to be shallower than the designed 70 nm from duplication, leading to the observation
that the experimental SPP is not coupled optimally to the incident light, and so the
reflectivity band is not as dark as the FEM prediction.
This band of low reflectivity is investigated further in figure 7.7. The reflected
intensity of light from a zigzag grating embedded in glass is illuminated with TE
polarised light, at a wavelength of 632.8 nm, and recorded as a function of the polar
angle of incidence,
θ
. At
θ
= 28
, a minimum is observed in the TE reflectivity from the
grating close to the first-order diffraction edge (calculated as
θ
= 18
), indicative of the
excitation of a SPP that has been scattered by one grating vector,
k
gx
. Incident light
resonantly couples to this diffracted SPP and is scattered back in to the specular order
with
π
phase retardation [
130
]. Out of phase with the specular reflection, a dark band is
observed with the missing energy being dissipated as Joule heating. For comparison, the
excitation of an in-plane SPP such as this on a mono-grating or bi-grating would require
101
7. Optical Response of Metallic Zigzag Bigratings
x (nm)
z (nm)
0
1
2
3
4
5
|E| × 10
8
(V/m)
0 100 200 300 400 500 600
-50 0 50 100 150 200 250
Figure 7.8: The
E
field plot of a SPP excited by TE radiation found in figure 7.7,
obtained by FEM. The colour scale shows the magnitude of the electric vector,
E
, for
a cross section of the zigzag grating, in the plane of SPP propagation (
k
to
ˆx
). The
arrows show the direction of the electric vector over space. The electric field phase
chosen for this travelling wave is arbitrary.
the light to be TM polarised. However, on this zigzag grating for a SPP Bragg scattered
by
k
gx
, excitation with TM polarised light is forbidden (see 7.3.1) and no minimum
in the TM data is observed. The data show excellent agreement with the theoretical
reflectivity obtained from finite element method (FEM) modelling, also shown in figure
7.7, with fitting parameters of
λ
632.8nm
=
17
.
5 + 0
.
55
i
, a groove depth of 29.9 nm. The
mark-to-space ratio of 0.75 in the model was determined from the electron micrographs
(figure 7.5(b)). These fitting parameters confirm that the grating was indeed shallowed
in the fabrication process. Surface roughness and small groove profile changes from the
embossing method are not accounted for in the modelling.
Using the fitted FEM model, the magnitude of the electric field in the
xz
plane is
plotted for this SPP mode in figure 7.8. The electric field is evanescently confined to
the surface with the field maximum occurring near the top, flat surface of the zigzags
at
z
=
d
= 29
.
9 nm and exponentially decaying away into both the air and the metal.
The SPP for this wavelength and angle propagates along the zigzag with a wavevector
given by
k
SP P
=
nk
0
sin θ k
gx
= 1
.
67
k
gx
, corresponding to a SPP wavelength of
λ
SP P
= 358 nm. The plane of intersection and temporal phase of the wave is arbitrarily
and chosen for the figure as
y
= 70 nm and 0
phase, respectively. The propagation of
the SPP along the zigzag, rather than over the grooves, has important consequences for
102
7. Optical Response of Metallic Zigzag Bigratings
.
.
in-plane wavevector, k
x
(m
1
)
.
angular frequency, ω (rad s
1
)
.
0.0
.
0.2
.
0.4
.
0.6
.
0.8
.
1.0
.
1
.
+2
.
0
.
+3
.
0
.
0.5
.
1
.
1.5
.
2
.
2.5
.
3
.
3.5
.
4
.
×10
7
.
×10
15
(a)
.
.
in-plane wavevector, k
x
(m
1
)
.
angular frequency, ω (rad s
1
)
.
0.0
.
0.2
.
0.4
.
0.6
.
0.8
.
1.0
.
1
.
+2
.
0
.
+3
.
0
.
0.5
.
1
.
1.5
.
2
.
2.5
.
3
.
3.5
.
4
.
×10
7
.
×10
15
(b)
Figure 7.9: (a) Experimental data and (b) numerically modelled results of the TE
polarised reflectivity as a function of in-plane wavevector and angular frequency, mapping
the SPP dispersion on a zigzag grating in glass. Blue lines show the positions of diffracted
light lines scattered by mk
gx
, where m = 1, +2, 0, +3.
the band structure for SPPs on this type of surface, and will be discussed in section 7.4.
It is desirable to fully confirm the coupling conditions outlined in equation 7.7
for other odd-order diffracted modes, rather than just the
±k
gx
scattered SPP. By
embedding a zigzag grating in a higher refractive index, such as glass, we are able to
access the +3k
gx
scattered SPP using visible frequency radiation. The experimentally
obtained dispersion of the SPPs on the glass/metal interface is mapped as a function of
in plane wavevector,
k
gx
, by varying the polar angle
θ
and angular frequency,
ω
, using a
polarised, collimated, monochromatic beam, produced by a white light source together
with a spectrometer, in the wavelength range 400
< λ
0
<
850 nm. The reflectivity of
incident TE polarised light at φ = 0
is shown in figure 7.9(a).
With the grating embedded in glass two Bragg-scattered SPP dispersion curves are
observed for this polarisation case. One originates at
k
gx
and the other at +3
k
gx
.
These odd-order diffracted SPPs are excited by TE polarised light. There is no evidence
of a SPP dispersion curve from a +2
k
gx
scattering event as, on a zigzag grating, it
is not excited by TE polarised light as predicted by equation 7.7. Using the fitted
parameters from figure 7.7, the dispersion plots were reproduced from FEM modelling
and found to be in good agreement and are shown in figure 7.9(b). The small difference
between the model asymptotic limits of the SPPs and experiment may be attributed to
103
7. Optical Response of Metallic Zigzag Bigratings
the differences between the dielectric function of silver used for the model [
84
] and our
experimental sample.
In summary, the SPPs are shown to be excited on the zigzag grating, with TE
polarised light coupling to the odd-order Bragg scattered SPPs. The dispersion of the
SPPs has been mapped for the two cases of the grating in air and in glass, and agrees
well with a numerical model. No even-order modes are observed when the sample is
illuminated with TE polarised light. There is no excitation of odd-order modes with
TM polarised light.
7.3.4 Transverse Magnetic Coupling
The observation of even-order Bragg scattered SPP modes coupled to by TM radiation
is expected from the allowed charge distributions examined in section 7.3.1. For our
sample, the experimental observation of this mode is difficult, as the zigzag grating is
not optimally coupled to the SPP due to shallowing of the grating during fabrication,
and the even-ordered diffraction is also very weak, only mediated by multiple scattering
processes of ±k
gx
.
To investigate the weak coupling of TM polarised light to the +2
k
gx
SPP, spectral
plots are analysed for a range of angles for which, theoretically, only the 2
k
gx
SPP can be
present. A zigzag grating in air was investigated over an angle range of (39
< θ <
55
),
to remove the ambiguity of which SPP mode is under observation. Comparison to the
TE reflectivity, which should not couple to the SPP, further helps identify the mode.
An example spectrum for both TE and TM polarised illumination at
θ
= 53
is shown
in figure 7.10. There is a clear broad absorption of the light in the TM case which is
missing from the TE case, reducing the reflectivity by approximately 7%.
Both the TE and TM reflectivity drop at higher frequencies due to the UV absorption
inherent in the optical response of silver. To remove this high frequency absorption
and better judge the position of the broad reflectivity discontinuity associated with
a weakly coupled SPP, the TE and TM reflectivity may be normalized with respect
to each other. The TE reflectivity spectra, containing no observed SPP modes but
containing the higher frequency absorption of silver, may be used to normalize the TM
data and remove the high frequency silver absorption. This allows a greater accuracy in
the determination of the broad TM coupled SPP mode’s position in frequency, which
is judged by the fitting of a Gaussian curve to the local maximum of the normalized
reflectivity in figure 7.11(a). To further confirm that this reflectivity anomaly is due
to the weak excitation of the +2
k
gx
scattered SPP, the dispersion of this mode in
frequency for the angle range 39
< θ <
55
is plotted on the modelled dispersion of
the 2
nd
order mode in figure 7.11(b), obtained from the same FEM model that showed
104
7. Optical Response of Metallic Zigzag Bigratings
angular frequency, ω (rad s
-1
)
reflectivity (a.u.)
2.5 3 3.5 4
× 10
15
0.8 0.85 0.9 0.95 1
R
TE
R
TM
Figure 7.10: An example experimental reflectivity plot for TE and TM polarised light
in the visible range at a fixed polar angle of
θ
= 53
. The broad absorption in the TM
case is attributed to the weak excitation of a +2k
gx
scattered SPP.
excellent agreement for the TE case in figure 7.6. The SPP mode dispersion is found to
be in good agreement with the theoretical prediction for the +2
k
gx
scattered SPP, and
we conclude that this 2
nd
order mode is excited by TM polarised radiation, and is not
excited by TE polarised light, in agreement with section 7.3.1.
Figure 7.12 shows the dispersion of the SPPs on the zigzag grating embedded in
glass and mapped by using TM polarised light, obtained both experimentally and
using FEM modelling. The only noticeable feature is a band of low reflectivity at high
frequency, which does not appear to be associated with any of the in-plane diffracted
light lines. This absorption of light is due to the excitation of a SPP that has been
scattered by an out-of-plane grating vector, (
k
gy
), and is observed as a flat parabolic
curve. The explanation of this SPP band’s shape for out-of-plane scattered SPPs
intersecting the plane of incidence can be found previously in chapter 2. Both the
experimental and theoretical figures show this band, although the experimental band
lies at a higher frequency compared to the theoretical curve, due to the difference in the
metal’s dielectric function between that of the experimental sample and the values used
105
7. Optical Response of Metallic Zigzag Bigratings
angular frequency, ω (rad s
-1
)
R
TE
R
TM
2.5 3 3.5 4
× 10
15
0.85 0.9 0.95 1
(a)
in-plane wavevector, k
x
(m
-1
)
angular frequency, ω (rad s
-1
)
0.75
0.80
0.85
0.90
+2
0
0.4 0.6 0.8 1 1.2
× 10
7
3.3 3.4 3.5 3.6
× 10
15
(b)
Figure 7.11: (a) An example spectral plot and fit at
θ
= 53
for the TM reflectivity
normalised to the TE reflectivity. (b) comparison between theory (greyscale) and
experiment (black points) for a range of angles (39
< θ < 55
).
.
.
in-plane wavevector, k
x
(m
1
)
.
angular frequency, ω (rad s
1
)
.
0.0
.
0.2
.
0.4
.
0.6
.
0.8
.
1.0
.
1
.
+2
.
0
.
+3
.
0
.
0.5
.
1
.
1.5
.
2
.
2.5
.
3
.
3.5
.
4
.
4.5
.
×10
7
.
×10
15
(a)
.
.
in-plane wavevector, k
x
(m
1
)
.
angular frequency, ω (rad s
1
)
.
0.0
.
0.2
.
0.4
.
0.6
.
0.8
.
1.0
.
1
.
+2
.
0
.
+3
.
0
.
0.5
.
1
.
1.5
.
2
.
2.5
.
3
.
3.5
.
4
.
4.5
.
×10
7
.
×10
15
(b)
Figure 7.12: Experimental data of TM polarised reflectivity as a function of in-plane
wavevector and angular frequency, mapping the SPP dispersion on a zigzag grating in
glass. Blue lines show the positions of diffracted light lines scattered by
mk
gx
, where
m = 1, +2, 0, +3.
106
7. Optical Response of Metallic Zigzag Bigratings
for FEM. In the experimental case, this frequency is at the limit of the monochromator’s
spectral resolution, and so the signal to noise ratio is lower than for previous plots.
This out-of-plane SPP mode lies significantly removed from the diffracted light line
for the
±k
gy
light cone, which is not plotted as it occurs at a higher frequency than the
displayed spectral range. Comparing this difference,
k
x
, for in-plane modes observed
under TE illumination, the observation of a
k
gy
scattered SPP mode with a large enough
k
y
to be observed suggests a high level of anisotropic dispersion of the SPP in the
k
x
and k
y
directions. This will be discussed further in section 7.5.
In summary, the observation of weak even-scattered SPPs excited by TM polarised
light has been demonstrated experimentally. To facilitate stronger coupling by even-
order scattered SPPs and TM polarised light, an even
k
gx
component may to be added
to the zigzag profile, but this would remove the mirror symmetry of the grating and the
polarisation selectivity would be destroyed. This will be discussed further in chapter
8. An out-of-plane SPP associated with
±k
gy
has also been observed, indicating that
the SPP dispersion in the
ˆ
y
direction is significantly perturbed from the in-plane (
ˆ
y
)
dispersion case. This anisotropic propagation of SPPs will be discussed further in section
7.5.
7.4 Band Structure of SPPs on a Zigzag Grating
The formation of SPP band-gaps relies on the ability for a propagating SPP mode to
Bragg scatter and interfere with its counter-propagating self [
35
,
85
]. This interference
forms two possible SPP standing waves on the surface of the grating, which will typically
have two possible field solutions, with the fields of one solution spatially shifted by
λ
gx
/
4
with respect to the other. Since only diffracted SPPs are observed in the light-cone,
the minimum momentum by which a SPP must scatter to intersect an equivalent
counter-propagating mode, and still be able to couple to light and be observed, is a
total of 2k
gx
.
On traditional surface-relief gratings, the positions of the standing wave anti-nodes
correspond to induced surface charge on the grating grooves. The two possible standing
waves will cause the surface charge density to sit in different arrangements per solution
which will, in general, sit charge in different potentials. This means that the energy
of the two standing wave solutions will differ, with no propagating SPPs between the
two energies due to destructive inference. This is the essence of plasmonic band-gap
formation.
On a zigzag grating, the SPPs propagating along the
k
gx
direction do not run
over surface relief grooves, but follow the zigzag shape along the surface. The two
possible standing waves which form when counter-propagating SPPs interact will differ
107
7. Optical Response of Metallic Zigzag Bigratings
Figure 7.13: An iso-frequency scattergram mapping
k
-space contours for a 600 nm
zigzag grating for an energy of 2.14 eV (
λ
0
= 580 nm). The blue circles indicate the
±k
gx
scattered light circles.
in energy by an amount determined by the zigzag structure of the surface and not
the shape or depth of the grooves. The positions of the standing wave anti-nodes
correspond to induced surface charge along the zigzag, and by consideration of these
charge arrangements, the band-gap character of zigzag gratings may be inferred.
7.4.1 Band-Gaps at k
x
= 0
For a zigzag grating with a periodicity of
λ
g
= 600 nm in air, the
±k
gx
scattered
SPPs meet at
k
x
= 0 within the visible frequency range. An experimentally obtained
iso-frequency contour for these modes is displayed in figure 7.13. The dark bands of low
reflectivity map the SPP mode positions. At
k
x
= 0 the +1
k
gx
and
1
k
gx
SPP contours
cross, and no significant perturbation of these contours is observed. No band-gap is
detected at the mode intersections. The lack of interaction of these modes along
k
x
= 0
is consistent with the known scattering properties of a zigzag grating. The grating
profile, being totally described by odd functions, contain no significant 2
k
gx
components.
The result of which is that a SPP must undergo multiple scattering processes of
±nk
gx
,
where
n
is odd, in order to diffract and intersect other SPP modes. These multiple
scattering processes are very weak, and result in such a small perturbation of the modes
as they intersect in k-space that no band-gap is observed experimentally.
108
7. Optical Response of Metallic Zigzag Bigratings
λ
gx
2
0 +
λ
gx
2
(a)
λ
gx
2
0 +
λ
gx
2
(b)
Figure 7.14: Cartoon of the two standing wave solutions for SPPs at
k
x
= 0, projected
onto the zigzag surface. Since the anti-nodes of the SPP standing waves lie on the
zigzag midsections in one case (a) and at the apexes in the other case (b), the standing
waves exist in different electromagnetic environments and a band-gap may form.
While no band-gap is measured, the existence of a band-gap in this case is not
forbidden. Figure 7.14 shows a diagrammatic representation of the expected locations of
nodes and anti-nodes at
k
x
= 0 for the
xy
plane. As previously mentioned, the SPPs on
zigzag gratings run along the zigzag, not over the grooves. For two SPP standing waves
to be energetically dissimilar, the pertinent consideration is the field arrangements along
the zigzag structure itself, which is why we consider the
xy
plane. The wavevector of
this standing wave is half that of the Bragg vector by which the SPPs have scattered,
in this case 2
k
gx
/
2, leading to a standing wave with the periodicity of
λ
gx
. It is seen
in the figure that the areas on which the surface charge is induced are at the zigzag
apexes in one case (figure 7.14(a)), and at their midsection in the other case (7.14(b)).
Using a FEM model, the electric field for these two possible standing waves at
k
x
= 0
are calculated plotted in figure 7.15.
The grating is modelled with the following parameters:
λ
gx
= 600 nm
, λ
gy
=
150 nm
, d
= 40 nm, with a mark-to-space ratio of 1 and a frequency-dependent silver
dielectric function [
84
]. We may identify if these standing waves differ in energy, and
which of these two is higher in energy relative to the other, by the field decay length into
the two media. The higher-energy solution, having been shifted higher in frequency and
therefore towards the diffracted light line, will be more photon-like than the lower energy
solution, its fields extending further into the dielectric than the low-energy solution. The
lower-energy solution will have been shifted lower in frequency, away from the light line
109
7. Optical Response of Metallic Zigzag Bigratings
x (nm)
y (nm)
0.1
0.2
0.3
0.4
0.5
|E| (V/m)
0 300 600
75 150
(a)
x (nm)
z (nm)
0.05
0.10
0.15
0.20
0.25
0.30
|E| (V/m)
0 100 300 500
0 100 200 300 400
(b)
x (nm)
y (nm)
0.1
0.2
0.3
0.4
0.5
|E| (V/m)
0 300 600
0 75 150
(c)
x (nm)
z (nm)
0.1
0.2
0.3
0.4
|E| (V/m)
0 100 300 500
0 100 200 300 400
(d)
Figure 7.15: The magnitude of electric field for the SPP standing waves at
k
x
= 0. (a-b)
The high-energy solution for (a) the
xy
plane at
z
= 41 nm and (b) the
xz
plane, for
y
= 75 nm. (c-d) The low-energy solution for (c) the
xy
plane at
z
= 41 nm and (d) the
xz plane, for y = 75 nm.
and so will be more plasmon-like, constrained closer to the surface. These deductions
can be applied in the comparison between the fields of the high energy standing wave in
figure 7.15(b) and the lower energy standing wave in figure 7.15(d). Subfigures 7.15(a)
and 7.15(c) show the magnitude of electric field for an arbitrary phase in a plane placed
1 nm above the zigzag surface at
z
= 41 nm. The two field distributions have the
expected period of
λ
gx
. The higher-energy standing wave (figure 7.15(a)) has the field
maxima along the edges of the zigzag grooves, while the lower energy standing wave
has the field hotspots on the zigzag apexes. It is this energy difference between the two
possible standing waves which leads to the plasmonic band-gap.
Analogous to the case of surface relief gratings, whereby the charge sits at the
110
7. Optical Response of Metallic Zigzag Bigratings
x (nm)
y (nm)
0 300 600
0 75 150
in-plane wavevector, k
x
(m
1
)
angular frequency, ω (rad s
1
)
-1.5 -1 -0.5 0 0.5 1 1.5
× 10
1.95 1.96 1.97 1.98 1.99 2
× 10
15
(a)
x (nm)
y (nm)
0 300 600
0 75 150
in-plane wavevector, k
x
(m
1
)
angular frequency, ω (rad s
1
)
-1.5 -1 -0.5 0 0.5 1 1.5
× 10
1.92 1.93 1.94 1.95 1.96 1.97
× 10
15
(b)
Figure 7.16: Manipulation of the band-gap observed at
k
x
= 0 through the increasing
of zigzag amplitude. The unit cell and corresponding SPP dispersion around the
intersection of the +1
k
gx
and
1
k
gx
scattered SPPs are shown for (a) The zigzag
grating presented in this chapter and (b) a theoretical zigzag with a larger zigzag
amplitude.
maxima or minima of the grooves (outlined in the background theory chapter 2), we
expect the dependence of this band-gap size to vary as some function of zigzag (in the
surface plane) amplitude.
An eigenmode solution is found for two possible unit-cells using FEM modelling,
the results of which are shown in figure 7.16. The parameters for this model used the
unit cells shown, with depths of 40 nm, silver parameters from literature [
84
] and with
the global environment refractive index
n
= 1
.
518. The large-amplitude zig-zag was
produced by shifting the central apex in the
y
direction by 150 nm. This causes multiple
zigzags to overlap in a single unit cell, in order to maintain the sub-wavelength period of
λ
gy
= 150 nm. It is clearly shown that with an increase of zigzag amplitude, a band-gap
is opened at
k
x
= 0. This gap is small, as the SPPs must still rely on multiple scattering
processes to diffract and interact. This is the first example in this thesis of using the
zigzag structure and symmetry to manipulate the band structure of the SPPs, a topic
111
7. Optical Response of Metallic Zigzag Bigratings
in-plane wavevector, k
x
(m
1
)
angular frequency ω ( rad s
1
)
BZ
-1
+2
0.51 0.52 0.53
× 10
7
2.8 2.82 2.84
× 10
15
Figure 7.17: Modelled SPP dispersion around the intersection point of the
1
k
gx
and
+2k
gx
scattered SPPs meeting at the first BZ boundary (red dashed line).
which will be examined further in chapter 8.
7.4.2 Band-Gaps at the First Brillouin Zone Boundary
Normally at Brillouin zone (BZ) boundaries, diffractive coupling results in two counter
propagating SPPs, which establish a standing wave [
85
]. The two standing wave solutions
correspond to different field distributions with respect to the grating profile, and between
these two energy solutions no surface modes propagate: a SPP band-gap forms. However,
the experimentally mapped dispersion shown in figures 7.6 and 7.9(a), and also the
predictions from FEM modelling (figures 7.6 and 7.9(b)) shows no measurable SPP
band-gaps at the first BZ boundaries for this zigzag grating.
There are two considerations to be made here to determine the possibility of a band-
gap between the
k
gx
and +2
k
gx
scattered SPPs: (1) is there a sufficient scattering
amplitude present for these modes to Bragg scatter and interact with one another?, and
(2) how do the possible standing wave states differ in energy to produce a frequency
band of disallowed SPP propagation, if at all?
Firstly, for the
k
gx
and +2
k
gx
SPPs modes to scatter and interact strongly, they
require a total momentum of 3
k
gx
. If the grating were to contain a 3
k
gx
component in
it’s surface profile, this would be a simple direct-scattering event and a strong interaction
can occur. Since we observe the 3
k
gx
scattered SPP in the reflectivity mapped in figure
7.9(a) (interacting with zero-order light), we can safely assume that the grating provides
112
7. Optical Response of Metallic Zigzag Bigratings
λ
gx
2
0 +
λ
gx
2
(a)
λ
gx
2
0 +
λ
gx
2
(b)
Figure 7.18: Cartoon of the two standing wave solutions for SPPs at the 1
st
BZ. Since
the peaks and troughs of the zigzag exist in equivalent electromagnetic environments,
the solutions are degenerate in energy, and no band-gap forms.
a strong 3
k
gx
scattering mechanism. However, despite the ability for the grating to
scatter SPP modes with a reasonable 3
k
gx
component, there is still no observable
band-gap between the
k
gx
and +2
k
gx
at the first BZ boundary. A numerical model
of these modes crossing at the BZ is shown in figure 7.17. The numerical model is
used to extract the possible eigenmodes of the SPPs at the 1st BZ boundary, with
modelling parameters of
λ
gx
= 600 nm
, λ
gy
= 150 nm
, d
= 40 nm
,
a mark-to-space ratio
of 1 and
ε
(
ω
) from literature [
84
]. No band-gap is observed, to within the accuracy of
the numerical model.
The lack of band-gap at the first BZ boundary is explained considering the allowed
standing-wave symmetries on a zigzag grating. Generally, the in-plane wavevector of
a standing SPP wave is half that of the total Bragg vector by which the two counter
propagating SPPs have been scattered. For the
k
gx
and +2
k
gx
scattered SPPs,
crossing at the first BZ boundary, the SPP wavevector is 3
k
gx
/
2. Through symmetry,
there are two possible solutions for this standing wave on a zigzag grating, with the
areas of high field of one solution shifted spatially by
λ
gx
/
4 with respect to the other
solution. These two solutions are drawn in figure 7.18, illustrating the node and anti-node
positions of a standing wave along the zigzag. Since SPPs run along the zigzag pattern,
(rather than over the grooves), the only possible manner by which to organise charge
in dissimilar energetic arrangements is to consider how the charge may accumulate
on the zigzag groove sides. Comparing this simple cartoon with the magnitude of
electric field in the
xy
plane, 1 nm above the grating surface, shown in figure 7.19, we
113
7. Optical Response of Metallic Zigzag Bigratings
x (nm)
y (nm)
0.1
0.2
0.3
0.4
|E| (V/m)
0 300 600
0 75 150
(a)
x (nm)
y (nm)
0.1
0.2
0.3
0.4
|E| (V/m)
0 300 600
0 75 150
(b)
Figure 7.19: The magnitude of electric field,
|E|
for the degenerate SPP standing waves
at the first BZ. (a) One solution for the
xy
plane at
z
= 41 nm and (b) The second
solution for the xy plane at z = 41 nm.
see that the unit cell contains three hotspots, corresponding to the expected period
of
λ
SP P
= 2
π/|k
SP P
|
= 2
λ
gx
/
3, with the field arrangements sitting in the locations
predicted by the cartoon. These field plots are extracted from the same numerical model
that mapped the dispersion in figure 7.17. In both cases, one hotspot sits on a zigzag
apex, and the other two sit symmetrically either side on the zigzag edges.
The electric field arrangements for the two possible standing wave modes at the BZ
boundary in the grooves (in the
xy
plane with
z
= 20 nm) and in the
xz
plane (with
y
= 75 nm), are extracted from the numerical model and plotted in figure 7.20. These
standing waves have three ‘hotspots’ per unit cell, corresponding to a standing wave
with the expected period of
λ
SP P
= 2
λ
gx
/
3. Comparing the magnitude of electric field
in the grooves for the
xz
plane in figures 7.20(b) and 7.20(d), it is clear that the field
arrangements are shifted spatially with respect to each other by
λ
gx
/
4, as one would
expect for two standing wave solutions which in general will lie
π/
2 degrees out-of-phase
with each other. Figures 7.20(a) and 7.20(c) show the calculated field arrangements as
figure 7.18 illustrated. Lying at the same potential, these field arrangements are clearly
equivalent, shifted spatially by
λ
gx
with respect to each other, yet both occupying a
degenerate electromagnetic environment. The magnitude of electric field in the
xz
plane
presented in figures 7.20(b) and 7.20(d) show that the decay of the electric fields into
both bounding media are the same for each case, and that neither solution can be
labelled more ‘photon-like’ nor ‘plasmon-like’ than the other. This further demonstrates
the equivalence in energy of these two standing waves.
The equivalence in energy of these two standing wave solutions can be broken by
114
7. Optical Response of Metallic Zigzag Bigratings
x (nm)
y (nm)
0.02
0.04
0.06
0.08
0.10
|E| (V/m)
0 300 600
0 75 150
(a)
x (nm)
z (nm)
0.02
0.04
0.06
0.08
0.10
0.12
0.14
|E| (V/m)
0 100 300 500
0 100 200 300 400
(b)
x (nm)
y (nm)
0.02
0.04
0.06
0.08
0.10
0.12
|E| (V/m)
0 300 600
0 75 150
(c)
x (nm)
z (nm)
0.05
0.10
0.15
|E| (V/m)
0 100 300 500
0 100 200 300 400
(d)
Figure 7.20: The magnitude of electric field for the degenerate SPP standing waves at
the first BZ. (a-b) One solution for the (a)
xy
plane at
z
= 20 nm and (b) the
xz
plane,
for
y
= 75 nm. (c-d) The second solution for the (c)
xy
plane at
z
= 20 nm and (d) the
xz plane, for y = 75 nm.
the removal of the zigzag grating’s mirror plane. By making the zigzag asymmetric,
the electric field arrangements for both possible standing waves differ in energy and a
band-gap may form. This is demonstrated in chapter 8
7.5 Anisotropic Propagation of SPP Modes for Self Colli-
mation
Plasmonic circuits couple and direct SPPs along designed structures to provide a transfer
of power and information [
55
,
131
]. These optoelectronic devices require the development
of SPP optics which behave like their classical optics counterparts.
115
7. Optical Response of Metallic Zigzag Bigratings
One of the most crucial roles of these surface-optics is the ability to generate
a collimated SPP beam, so that the SPPs may be directed efficiently without any
detrimental loss across the device due to divergence. The ideal device should receive
incident SPPs over a wide range of angles and the SPPs should then propagate through
the component with a narrow angular divergence. This effect has been observed for
light in photonic crystals, and was named self-collimation. [132, 133]
Recently, such components have been manufactured that use square bi-gratings to
manipulate the local curvature of the SPP iso-frequency contour. These gratings have
demonstrated self-collimation in the microwave regime [
134
] as well as in the visible
[
61
]. They have also been designed to achieve all-angle negative refraction of surface
waves [
135
,
136
]. The work by Stein et al. [
61
] showed that the self-collimation effect
could be achieved around the M symmetry point in
k
-space due to the mini-gaps formed
by band-gapping SPPs over a momentum range of 1
µ
m
1
. It is desirable that this
momentum range is large, so as to maximise the angular divergence over which the
device could collimate SPPs.
We have found that zigzag gratings possess a mechanism for the collimation of SPPs
at a single wavelength. This mechanism is similar to previous work, as it relies on
plasmonic band-gaps to cause anisotropic propagation of SPPs along the surface, which
is observed as a deformation of the SPP iso-frequency contours to approximately flat
bands with no curvature. However, the azimuthal angle range over which SPPs possess
a single direction is far larger than previously reported results, extending over the entire
observable region of k-space.
Up until now, we have only considered the effects of the
λ
gx
periodicity of the zigzag
grating with respect to the SPP propagation. With a designed orthogonal pitch of
λ
gy
= 150 nm, the diffracted SPPs from a
k
gy
scattering mechanism have such a large
momentum that for the plane of incidence
φ
= 0
, the SPP features will lie well above
the visible frequency range.
However, an important point to note about SPPs travelling along the
k
gy
direction
is that they will be travelling over surface relief grooves and not, like the
±k
gx
surface
plasmons, along the zigzag contours. This means that the charge arrangements for
standing-waves at the BZ boundaries will differ in energy considerably, as for traditional
surface-relief gratings. Plasmonic band-gaps in the
k
gy
direction are expected, and
will be similar to those on typical bigratings. Evidence for these band-gaps are seen in
figure 7.21. The experimental results show the dispersion mapped using the reflectivity
of the zigzag grating to indicate the SPP mode positions for a wavelength range of
450
< λ
0
<
800 nm and an angular range 7
< θ <
65
, for two planes of incidence. One
plane of incidence includes the
k
gx
grating vector (
φ
= 0
) and the orthogonal plane of
incidence (
φ
= 90
) contains the
k
gy
vector. In both cases the
±
1
k
gx
scattered SPP is
116
7. Optical Response of Metallic Zigzag Bigratings
in-plane wavevector, k
x
(m
-1
)
angular frequency, ω (rad s
-1
)
0.0
0.2
0.4
0.6
0.8
1.0
-1
+1
+2
0
0 0.2 0.4 0.6 0.8 1 1.2
× 10
7
2.5 3 3.5 4
× 10
15
(a) R
ss
at φ = 0
in-plane wavevector, k
y
(m
-1
)
angular frequency, ω (rad s
-1
)
0.0
0.2
0.4
0.6
0.8
1.0
+1
0
0 0.2 0.4 0.6 0.8 1 1.2
× 10
7
2.5 3 3.5 4
× 10
15
(b) R
pp
at φ = 90
Figure 7.21: Experimentally obtained dispersion plots for a zigzag grating at
φ
= 0
and φ = 90
observed. For
φ
= 0
the SPP dispersion is an asymptotic curve, while for
φ
= 90
it
is seen as a hyperbolic conic intersection, but both map the same SPP in momentum
space. For the coupling of light to the SPP in figure 7.21, the polarisation of the incident
light has been rotated 90
, and so is TM polarised.
It is found that in the
φ
= 90
orientation, the SPP dispersion curve asymptotes far
faster than the
φ
= 0
case. This lower asymptote is evidence of a large band-gap at
the first BZ in the
k
gy
direction, with the mode meeting the BZ boundary with zero
group velocity at 2
.
1
×
10
7
m
1
. This band-gap is the result of the interaction between
the
±k
gx
and
±k
gx
+
k
gy
scattered SPPs which are separated by a single scattering
vector that is also a harmonic of the grating surface profile,
k
gy
. Strong interaction is
then expected, and the freedom of the charge to organise into energetically dissimilar
arrangement in the grooves leads to a large plasmonic band-gap in this direction.
The relevance of this large band-gap to the collimation of SPPs becomes apparent
in the evolution in wavelength of the SPP iso-frequency contours mapped using imaging
spectrometry, shown in figure 7.22.
The scattergrams map the iso-frequency contours of the
±
1
k
gx
scattered SPPs
in
k
-space, with the minimum of reflection indicating the SPP mode position as the
reflection is suppressed by SPP excitation. The polarisation is set so that the electric
field lies orthogonal to the
k
gx
vector at
k
y
= 0, satisfying the TE excitation condition
for these SPPs in this plane. As detailed in section 7.3.3, the
±
2
k
gx
scattered SPPs are
not excited with this polarisation, and are not observed. The grating used for these
scattergrams is a silver zigzag grating in air as detailed in section 7.3.3, with a depth of
d 40 nm.
117
7. Optical Response of Metallic Zigzag Bigratings
(a) 650 nm (b) 600 nm (c) 580 nm
(d) 550 nm (e) 500 nm (f) 450 nm
Figure 7.22: Measured iso-frequency contours of a zigzag grating for a range of wavelengths. The blue circles indicate calculated
diffraction edges.
118
7. Optical Response of Metallic Zigzag Bigratings
Figure 7.23: The SPP iso-frequency contours mapped using imaging scatterometery at
λ
0
= 450 nm. The green arrows show the direction of the SPP group velocity along
the contour, indicating that for this wavelength the SPP waves only travel in the
±k
gx
direction. The red dotted lines show the position of the BZ boundary.
As the frequency increases (the wavelength shifting from red to blue), the
±
1
k
gx
scattered SPP contours grow increasingly flat-banded. This is due to the increasing
overlap and interaction with the
±
1
k
gx
±
1
k
gy
scattered SPPs and the formation of
a large band-gap well outside the available momentum space to the free-space light.
At
λ
0
= 450 nm, the SPP bands become flat, with essentially no curvature. This
scattergram is repeated in figure 7.23, with some additional important annotations.
Since the group velocity of the SPP wave is determined from
v
g
=
ω
(
k
), the
direction of the SPP energy flow is in the direction orthogonal to the SPP contour (see
chapter 6 for more details on this inference). Six example group velocity directions are
shown as green arrows in figure 7.23, showing that, for a given scattered SPP, the group
velocity is only in the
k
x
direction. This corresponds to a self-collimated SPP wave
across a momentum range of at least 2
.
2
×
10
7
m
1
, an order of magnitude higher than
previous results [
61
], and successfully collimates SPPs for an incident azimuthal range
of 63.4
< φ < 63.4
at λ
0
= 450 nm.
An Important observation is to be made of figure 7.23; that the
±
1
k
gx
scattered
SPP contours lie outside of the BZ boundary (red dotted lines). Two comments must be
made about this: the first is that the BZ represents the smallest unit cell which, when
repeated by translational symmetry operations, reproduces fully the band structure of
the system. Clearly, the BZ lying between the two red lines does not replicate the band
structure of the grating once translated. This is because the
±
2
k
gx
scattered SPPs
119
7. Optical Response of Metallic Zigzag Bigratings
are not coupled strongly to light in our experiment. The eignestates for the SPPs are
still present in the BZ, just not excited by the incident radiation, and so the validity of
the BZ holds. The second point is that the
±
1
k
gx
SPPs have passed through the BZ
boundary unperturbed. This is important, as if the contours were perturbed at this
boundary, the efficiency of the collimation of SPP waves would be impacted. Because
band-gaps at the first BZ are forbidden (not just weak) in the
k
x
direction, unwanted
perturbation of the SPP contour at this boundary will not occur. This is a condition
due to the mirror symmetry of the zigzag grating surface, and an analogy cannot be
found in traditional gratings.
7.6 Conclusions
This chapter has introduced a new type of diffraction grating, a zigzag grating. This
grating uses sub-wavelength surface structure to provide a diffractive periodicity to
wavelength-scale light. SPPs may be diffractively coupled to using a metallic zigzag
grating, and their excitation and band structure are found to depend on the symmetry
of the zigzag pattern.
The polarisation of light coupling to SPPs on such a grating is found to be dependent
on the diffracted order used. For odd-ordered diffraction, TE polarised light couples to
the SPPs, while for even-ordered diffraction, TM polarised light is required. This has
been explained using a simple analytical formula which considers the available normal
component of electric field to the surface of the grating, and the polarisation selectivity
has been demonstrated experimentally on fabricated silver zigzag gratings.
This polarisation selectivity may be found to be of use for plasmonic devices in
which polarisation separation is desirable. As an example, the incorporation of zigzag
gratings in metal-insulator-metal structures designed for the generation of light could
yield a polarisation dispersing light source. These SPP mediated light sources, when
incorporated with a zigzag grating, would emit different polarisations of light in to
different diffracted orders. Breaking the mirror symmetry of the zigzag pattern provides
a route to polarisation-independent absorption of light into SPPs, a topic which is
explored in greater detail in chapter 8.
The band structure of SPPs on zigzag gratings is found to be highly dependent on
the surface symmetry. Most strikingly, the formation of a band-gap at the first BZ
boundary is forbidden by the degeneracy of the allowed standing wave states.
Using scatterometry, the propagation of SPPs on such surface is shown to be highly
anisotropic, due to the large band-gaps which occur orthogonal to the diffraction plane.
These highly perturbed SPP contours, combined with the forbidden band-gaps at
the first BZ boundary in the
x
-direction, lead to wide-angle surface wave collimation.
120
7. Optical Response of Metallic Zigzag Bigratings
The use of zigzag gratings to generate surface-waves with highly directional planar
wave-fronts in plasmonic circuits, over a wide range of azimuthal angles, could be of
great interest to optoelectronic engineers. Further investigation of this phenomenon is
left for future work, with the recommendation of using near-field imaging of SPPs to
characterise the surface waves and exploring also the collimating effect of the grating.
121
Chapter 8
Optical Response of Asymmetric
Zigzag Bigratings
8.1 Introduction
This chapter demonstrates how the surface symmetry of a zigzag grating may be used
to manipulate both the coupling of light to SPPs, and the band structure of SPPs on
the zigzag surface. Much of the discussion in this chapter draws parallels with the
zigzag grating explored in chapter 7. The zigzag grating explored in chapter 7 will be
referred to as a ‘symmetric’ zigzag grating, due to the presence of mirror symmetry in
the
yz
plane. The zigzag grating which shall be the topic of this chapter has no such
symmetry, and so will be referred to as an ‘asymmetric’ zigzag grating. The asymmetric
zigzag grating is formed from a set of sub-wavelength (non-diffracting) grooves that are
zigzagged along their length such that the zigzag apexes do not lie at high symmetry
points within the rectangular unit cell. The length scale of the zigzag pitch is on the
order of the wavelength of impinging light, so that the grating may diffractively couple
to SPPs.
The resulting structure is 2D chiral, much like ‘fish-scale’ nanowires reported previ-
ously, [
137
141
] but we restrict our investigation to the excitation and band structure
of SPPs along such a surface, rather than any optical chirality exhibited by the grating.
In this chapter it is demonstrated that any polarisation of light may couple to the
SPP modes on a zigzag grating which possesses no mirror symmetry. Furthermore, it
is shown that light of different polarisation will couple to the exact same SPP modes,
and these SPP modes propagate in the same direction, regardless of polarisation. This
latter point is subtle, but important, as previous work has shown that the excitation
of SPPs on crossed bigratings may also couple arbitrary polarised light into SPPs on
the surface [
44
,
77
,
142
], but in this case different incident polarisations states excite
122
8. Optical Response of Asymmetric Zigzag Bigratings
different, or multiple, SPPs. The coupling of arbitrarily polarised light to SPPs on deep
lamella gratings has also been shown [
143
], where the SPP modes evolve to become
similar in character to localised cavity modes. In the zigzag case presented here, light
of any arbitrary polarisation is coupled to the same SPP modes, and the energy flow
across the surface is mediated by these SPPs travelling in a single direction only. This
makes these asymmetric gratings an excellent possible component for efficient plasmonic
circuitry [131].
The manipulation of available Fourier components by altering the geometry of the
asymmetric zigzag surface also allows the design of the SPP band structure on such
gratings. We find in section 8.4, theoretically and experimentally, how the asymmetric
zigzag pattern can cause band gaps to form between the +
k
gx
and
k
gx
scattered SPPs,
by providing a direct scattering mechanism by which they may interact. We also show
in section 8.4 how the degeneracy of standing wave solutions at the first BZ boundary,
described in chapter 7 for the symmetric case, is destroyed in the asymmetric zigzag
case. The coupling of light to the two energetically dissimilar standing wave states is
also investigated, as the high energy solution is found to be poorly coupled to light
compared to the lower energy mode.
Finally, highly anisotropic propagation of SPPs along zigzag gratings, when combined
with the possible large band gap at the first BZ boundary, causes the formation of a
full-plasmonic band gap, for which propagation of SPP for a given frequency range is
forbidden in all directions. Full plasmonic band-gaps have shown potential in surface
plasmon based lasers [
47
49
,
144
], and are usually only found in systems with hexagonal
[114], not rectangular, symmetry.
8.2 The Asymmetric Zigzag Grating
The coordinate system used for the asymmetric zigzag grating is shown in figure 8.1.
Light impinges on the surface at a polar angle
θ
in the plane of incidence that lies
at an angle
φ
, with
φ
= 0
containing the primary grating vector,
k
gx
= 2
π
ˆ
x
gx
.
The polarisation of the light is defined as TM polarised light when the electric vector
is in the plane of incidence, and TE polarised light when the electric vector lies
perpendicular to the plane of incidence. The offset of the zigzag apex,
δ
is the distance
between the centre of the unit cell and the closest apex, as shown. It is this offset
that removes the mirror symmetry of the zigzag grating. The grating grooves are of
a depth
d
. The target parameters (for fabrication) used in this chapter are as follows:
λ
gx
= 600 nm, λ
gy
= 150 nm, d 40 nm
and
δ = 150 nm
with the grating made of silver.
The zigzag amplitude is defined as the distance between a low and high apex of the
zigzag divided by two.
123
8. Optical Response of Asymmetric Zigzag Bigratings
δ
φ
θ
E
T M
E
T E
d
x
y
z
λ
gx
λ
gy
plane of incidence
Figure 8.1: Coordinate system for the asymmetric zigzag grating. The experimental
parameters were λ
gx
= 600 nm, λ
gy
= 150 nm, d 40 nm, and δ = 150 nm .
The zigzag grating presented in this chapter has a reduced symmetry from the zigzag
grating presented in chapter 7. This is achieved by offsetting one of the zigzag apexes
along
ˆ
x
with respect to the centre of the unit cell by a distance
δ
= 150 nm, as shown in
figure 8.2. This produces a grating structure with no mirror or rotational symmetry in
real space, but still having a rectangular unit cell and a rectangular lattice in reciprocal
space. For ease of description, we refer to the ‘left hand side’ of the grating, the region
0 nm < x < 450 nm
(to the left of the apex), as ‘region 1’ and the right hand side of
between 450 nm < x < 600 nm as ‘region 2’.
The samples are produced using electron beam lithography as detailed in chapter 4.
Scanning electron micrographs of the silicon master and the template stripped sample
in silver are shown in figure 8.3, with the following parameters:
λ
gx
= 597
±
5 nm
, λ
gy
=
156 ± 9 nm, d 40 nm, δ = 126 ± 5 nm with a zigzag amplitude of 121 ± 3 nm.
Due to the template stripping method of fabrication, the produced silver grating
is an inverse duplicate of the grating master. The master SEM 8.3 also shows a clear
stitching error on the right of the SEM, where the zigzag continuity has been broken.
These stitching errors are infrequent and, over the area of the grating, do not effect the
optical response greatly.
Notice that the zigzag amplitude of the sample is greater than that of the unit
cell in figure 8.2. The experimental and modelled results will show that since an
124
8. Optical Response of Asymmetric Zigzag Bigratings
x (nm)
y (nm)
0 300 600
0 75 150
(a) Symmetric
x (nm)
y (nm)
0 300 600
0 75 150
0 150
region 1 region 2
300 + δ
(b) Asymmetric
Figure 8.2: Two possible unit cells for a zigzag grating. (a) The grating unit cell
explored previously in chapter 7. (b) An asymmetrical zigzag grating, with the central
apex of the zigzag shifted by 150 nm. This shift removes the mirror-symmetry of the
zigzag.
(a) Master (b) Metal grating replica
Figure 8.3: Scanning electron micrographs of (a) The asymmetric zigzag silicon master
and (b) the template stripped sample in silver.
125
8. Optical Response of Asymmetric Zigzag Bigratings
increased zigzag amplitude alone does not alter the symmetry of the zigzag surface, the
fundamental optical response of the grating is unaltered. For simplicity in understanding
only consideration of the unit cell in figure 8.2 is necessary to gain an insight into the
optical response of our sample.
8.3 The Coupling of Polarised Light to SPPs on an Asym-
metric Zigzag Grating
8.3.1 Theory
In chapter 7, it was shown that a zigzag grating possesses a surface profile that allows
surface charge density oscillations to be induced by either TE or TM polarised light. This
is because both TE and TM light provide a component of the incident electric field that
lies normal to some part of the metal surface. For the case of the grating examined in
chapter 7, the field profiles of these induced charge density oscillations were constrained
by the symmetry of the surface so that the wavevector of SPPs coupling to TE polarised
light are only the odd-orders of diffraction, while the TM polarised light could only
match the wavevector of even-order diffracted SPPs. These coupling conditions can
be deduced from the plane wave expansion equations 7.4 and 7.5. Since the functions
E
T E
and
E
T M
will possess the same mirror symmetry as the surface function, so the
Fourier expansions of
E
T E
and
E
T M
only require every-other Fourier coefficient to fully
describe the functions. This is what leads to the polarisation selectivity of this type of
grating.
However, by removing the mirror symmetry of the zigzag surface the equivalent
expressions for
E
T E
and
E
T M
in the asymmetric zigzag case will themselves not be
mirror symmetric. We can demonstrate this simply by considering the E field for normal
incident light, as shown diagrammatically in figure 8.4.
In this figure, the possible electric fields across the zigzag grooves are shown for
two electric vectors, one in the
x
direction (left), and one in the
y
direction (right).
The electric vectors pointing along a single axes like this represents two orthogonal
polarisations at a single instant in time when the light in incident normal to the surface.
The plane of incidence at
θ
= 0
is not defined, as the wavevector possesses no component
in the surface plane
xy
, but for
φ
= 0
and for a very small polar angle
θ
, the polarisation
is defined as TE for the vector lying along the
x
axis (right) and TM for the vector lying
along the
y
axis (left), so we shall refer to the left-hand diagram as the TE polarised
case, and the right-hand side as the TM polarised case.
The figure shows that for the electric vector oriented along the
x
or the
y
axis, the
incident field may provide a normal component to the grating surface and so induce
126
8. Optical Response of Asymmetric Zigzag Bigratings
E
T M
E
T E
E
x
0 0
E
x
Figure 8.4: (above) A schematic of the electric field lines in the grooves (black arrows) of
an asymmetric zigzag grating for two polarisation cases, and the resulting
E
x
component
(red arrows). The points connecting the apexes lead to fixed points of zero
E
x
, leading
to an asymmetric field distribution along the plane of propagation (bottom).
surface charge. Because of the surface geometry, charge density oscillations may be
induced on the grating surface and so there exists a possibility of exciting SPPs. Since
in our experiments
λ
gy
is sub-wavelength, we need only consider the electric field profile
along the
x
axis, as this is the axis along which momentum conservation requires
diffracted SPPs to propagate. The
x
-components of the electric fields in the grooves
are shown as red arrows in the figure. Possible example functions for each polarisation
which fulfil the asymmetry criteria of the grating are shown in the lower half of figure
8.4, which have been deduced from the diagrams above. These functions are neither odd
nor even, both possessing only the symmetry of the grating (a rotation and translation
operation or ‘glide operation’), and will in general vary in amplitude in the two regions.
The key point illustrated in this figure is that the normal component of electric field
varies in the
x
-direction asymmetrically with respect to the unit cell. This is because the
x
-component of the electric field originating from charge carriers must be pinned to zero
at the zigzag apexes by the geometry of the grating, and the position of these apexes
do not lie at points of any high-symmetry. The plane wave expansion description of
such general diffracted SPPs fields on such a surface will necessarily contain all Fourier
127
8. Optical Response of Asymmetric Zigzag Bigratings
components including both odd and even harmonics, for both polarisation cases.
In chapter 7, we calculated the Fourier series of
E
T E
(
E
T E
x
) and
E
T M
(
E
T M
x
)
by first deriving the electric field from the expected induced surface charge along
x
as
the functions
E
T E
x
and
E
T M
x
. Examination of the plane wave expansion of these fields
determines the coupling of light to different orders of diffracted SPP. In the asymmetric
case, the strict derivation of
E
T E
x
and
E
T M
x
is difficult, due to the lack of symmetry
in the grating’s geometry. We can instead construct an example function from careful
consideration of the conditions an
E
x
field must possess along such a grating. We place
five constraints on any function we choose: (1) That the function must be continuous
along
x
, with no sharp discontinuities or undefined points; (2) that the function is
periodic, with a period equal to that of a single grating period,
λ
gx
; (3) that the function
must equal
E
x
= 0 at the
x
coordinates which correspond to a zigzag apex; (4) that
between these zeros, the magnitude of
E
x
will reach a maximum value determined by
the polarisation and local geometry and; (5) the resulting function must possess the
same symmetry as the grating (a single C
2
rotation operation).
Such a function must be defined piecewise, with the same symmetry as the intuitive
example functions drawn in figure 8.5. Choosing a simple sine wave for each region, the
function we shall use is defined as,
E
T M
x
(x) =
E
T M
x
1
(1 + sin (
L
2
+
2πx
(L+δ)
)) 0 < x < L + δ
E
T M
x
2
(1 + sin (
L
2
+
2π(xLδ)
(Lδ)
)) L + δ < x < 2L
(8.1)
E
T E
x
(x) =
E
T E
x
1
(1 + sin (
L
2
+
2πx
(L+δ)
)) 0 < x < L + δ
E
T E
x
2
(1 + sin (
L
2
+
2π(xLδ)
(Lδ)
)) L + δ < x < 2L
(8.2)
Where 2
L
=
λ
gx
. The variable
δ
represents a positive offset of the central zero point
from
x
=
L
and the function is defined as periodic with a period 2
L
. These functions
are plotted in figure 8.5
The magnitude of
E
x
in the two regions (
E
T E
x
1
, E
T E
x
2
, E
T M
x
1
, E
T M
x
2
) will not in
general be equal and will vary as a function of the offset
δ
. This magnitude will be a
result of the dot product between the impinging field vector and the local surface normal,
which will be different for TE and TM polarised light. To a good approximation, the
amount of induced surface charge density will be proportional to this dot product, and
the resulting SPP electric field may be decomposed along the
x
axis to see which plane
wave components will couple strongly to it. Using these considerations the expressions
128
8. Optical Response of Asymmetric Zigzag Bigratings
-1.0 -0.5 0.0 0.5 1.0
x
E
x
0 L 2L
L L + δ
E
T E
x
E
T M
x
Figure 8.5: (a) The piecewise function representing
E
x
for TE (
E
T E
x
) polarisation (black
line) and TM (
E
T M
x
) polarisation (red dotted line). Blue background shows the zigzag
grating in the
xy
plane, with the
x
-coordinates aligned with the plot.
E
0
= 1 in this
diagram.
for maximum |E
x
| will be given by,
E
T E
x
1
= E
0
cos
tan
1
h
L δ

sin
tan
1
h
L δ

(8.3)
E
T M
x
1
= E
0
sin
2
tan
1
h
L δ

(8.4)
E
T E
x
2
= E
0
cos
tan
1
h
L + δ

sin
tan
1
h
L + δ

(8.5)
E
T M
x
2
= E
0
sin
2
tan
1
h
L + δ

(8.6)
where
h
is half the zigzag amplitude and
E
0
is the initial field magnitude. These
functions are plotted in figure 8.6 as a function of
δ
, and show how the
E
x
component
varies rapidly for region 2, as the increasing offset forces the normal component of the
surface increasingly in the
x
direction. In the extreme case of
δ
=
L
, TM polarised
light will be perpendicular to the region 2 surface normal, and TE will be parallel to it,
leading to E
T M
x
2
= E
0
and E
T E
x
2
= 0.
We may now expand these two functions as a Fourier sum to determine which
diffracted SPP wavevectors may be matched with incident plane waves for each polari-
sation case. We do this for an asymmetric zigzag grating with
δ
= 0
.
42
L
= 126 nm the
129
8. Optical Response of Asymmetric Zigzag Bigratings
0.0 0.2 0.4 0.6 0.8 1.0
δ
E
x
0 L/2 L
E
T M
0
E
T E
0
region 1
region 2
Figure 8.6: Magnitude of the electric field in region 1 (dashed) and region 2 (dotted)
for the two polarisation cases of TM (black) and TE (red) polarised light, as a function
of offset, δ.
offset which was measured from the SEMs of the produced sample. Since the
E
T E
x
and
E
T M
x
functions are neither odd nor even, we expand them as a complex Fourier sum,
and the take the magnitude of the Fourier components squared to obtain a qualitative
understanding of the relative scattering efficiencies [
116
]. The complex Fourier series
coefficients are given by,
C
n
=
1
L
Z
L
L
f(x) exp (inx) dx (8.7)
Where
n
is the order of the Fourier harmonic. The Fourier components for both
E
T E
x
and
E
T M
x
functions are plotted in figure 8.7. The efficiency with which light will couple
to the SPPs on an asymmetric zigzag grating will be directly linked to the amplitudes
of these Fourier coefficients, as the strength of each harmonic equates to the scattering
amplitude of the SPPs. These coefficients are different for each polarisation, but they
will not be zero.
This leads to the conclusion that both TE and TM polarised light will couple to
all orders of SPPs on such a grating, with different (but non-zero) coupling strength
depending on the diffracted order used to couple to the SPP. It is possible to excite any
diffracted SPP mode with any polarisation of light on such a grating. That is to say,
the same SPP mode may be driven with either polarisation.
130
8. Optical Response of Asymmetric Zigzag Bigratings
0.00 0.02 0.04 0.06 0.08
n
|C
n
|
2
-4 -3 -2 -1 0 1 2 3 4
(a) TE
0.002 0.006 0.010
n
|C
n
|
2
-4 -3 -2 -1 0 1 2 3 4
(b) TM
Figure 8.7: The square magnitude of the Fourier coefficients for
n
= 0
... ±
4 with
an offset of
δ
= 0
.
42
L
, which corresponds to the experimentally measured offset of
126 ± 5 nm for the (a) TE case and (b) TM case.
Incidentally, examining the Fourier coefficients for no offset (setting
δ
= 0), we
recover the polarisation selectivity predicted in chapter 7 for a symmetric zigzag grating.
8.3.2 Results
The coupling of light to SPPs on an asymmetric zigzag grating is demonstrated ex-
perimentally in figure 8.8. This figure plots reflectivity for both TE and TM po-
larised light illuminating an asymmetric zigzag grating as a function of polar angle
of incidence (
θ
), at
φ
= 0 and
λ
0
= 500 nm. The reflectivity was obtained using
the experimental method outlined in chapter 4, with a robotic table controlling the
polar angle and a spectrometer setting the incident wavelength. The polar angle
range is 7
< θ <
60
. The zigzag grating sample has the following parameters:
λ
gx
= 597
±
5 nm
, λ
gy
= 156
±
9 nm
, d
40 nm
, δ
= 126
±
5 nm with a zigzag amplitude
of 121 ± 3 nm, measured from the sample SEMs.
Both polarisations show a reflectivity minimum at
θ
19
, characteristic of the light
resonantly driving the
1
k
gx
scattered SPP. Both curves also show a second resonant
minimum at
θ
32
, indicative of the light interacting with the +2
k
gx
SPP. In both
the TE and TM case, these reflectivity minima lie at the same angle to within the
experimental error (shown by grey bars in the figure) for each polarisation, suggesting
it is indeed the same SPP mode being excited in each case.
131
8. Optical Response of Asymmetric Zigzag Bigratings
10 20 30 40 50
0.4 0.5 0.6 0.7 0.8 0.9 1.0
polar angle, θ
R
1 +2
Figure 8.8: The TE (black) and TM (red) reflectivity of an asymmetric zigzag grating
at a wavelength of
λ
0
= 500 nm. The blue dotted lines show the calculated position
of the
1 and +2 diffraction edges. The grey bars are a width equal to the minimum
step-size of the experiment, centred at the average mode minima.
This is an example of a zigzag grating coupling either polarisation to the SPP on the
zigzag surface, as the impinging electric field on the asymmetric zigzag profile provides
a normal component with which to induce local surface charge regardless of polarisation.
Since the electric vectors for TE and TM polarised light lie orthogonal to each other,
we may generalize and state that any plane polarisation of light will excite SPPs on
asymmetric zigzag gratings.
The reflectivity minimum associated with the +2
k
gx
scattered SPP at
θ
32
is
lower for the TM polarised case than the TE. While the minima associated with
1
k
gx
shows the opposite, with TM polarised light coupling stronger than the TE polarised
light. Returning to the Fourier coefficients shown in figure 8.7, this is unsurprising.
These coefficients relate to the strength by which surface fields may couple to the plane
wave light. With
|C
1
|
2
larger for the TE case, and
|C
2
|
2
larger for the TM case, we
expect the
±
1
k
gx
scattered SPPs to be couple more strongly to the TE polarised light,
and the
±
2
k
gx
scattered SPPs to be couple more strongly to TM polarised light, as
observed in our experimental results.
The dispersion of the in-plane SPPs (
φ
= 0
) propagating along an asymmetric
zigzag grating was measured from the reflectivity of TE and TM polarised light, and
these results are shown in figure 8.9. These data were obtained using the methods
132
8. Optical Response of Asymmetric Zigzag Bigratings
in-plane wavevector, k
x
(m
1
)
angular frequency, ω (rad s
1
)
0.2
0.4
0.6
0.8
1.0
0.2 0.4 0.6 0.8 1 1.2
× 10
7
2.5 3 3.5 4 4.5
× 10
15
+1
-1
0
+2
(a) TE
in-plane wavevector, k
x
(m
1
)
angular frequency, ω (rad s
1
)
0.2
0.4
0.6
0.8
1.0
0.2 0.4 0.6 0.8 1 1.2
× 10
7
2.5 3 3.5 4 4.5
× 10
15
+1
-1
0
+2
(b) TM
Figure 8.9: The dispersion of the SPP modes mapped as a function of (a) TE and (b)
TM reflectivity on an asymmetric zigzag grating. The blue lines are the calculated
positions of the diffracted light lines. The colour scale indicates the absolute reflectivity.
outlined in chapter 4, with the polar angle range of 7
< θ <
60
in steps of 0
.
5
, and a
wavelength range of 400 nm < λ
0
< 850 nm in steps of 2 nm.
The SPPs associated with
±
1
k
gx
and +2
k
gx
diffraction are clearly coupled to in
each case. In the TE case in figure 8.9(a), the
±
1
k
gx
SPPs are coupled strongly,
and the +2
k
gx
scattered SPP is coupled, albeit weaker than the
±
1
k
gx
case. Under
TM polarised illumination, the dispersion shown in figure 8.9(b) shows the opposite
response, with the stronger coupled mode having scattered by +2
k
gx
, and the weaker,
yet still coupled mode, having scattered by
±
1
k
gx
. This is consistent with the predicted
magnitude of the Fourier components for each polarisation case, shown in figure 8.7,
and may be considered a perturbation of the symmetric zigzag case detailed in chapter
7, with the polarisation response of the surface still partially governed by the same
zigzag considerations, but with additional, smaller Fourier components destroying the
symmetry which led to the perfect separation of TE and TM coupled SPPs. The
additional Fourier components in the surface profile have provided direct scattering
routes for the excitation of the +2
k
gx
scattered SPPs observed for both polarisation
cases in figure 8.9.
For unpolarised light, the dispersion is mapped in figure 8.10. The coupling to the
SPP dispersion is now found to be reasonably symmetric around the first BZ boundary
(at
5
.
5
×
10
7
m
1
), as the SPP eigenmodes of the surface are coupled to by either
the TE or TM components of the unpolarised light. The level of absorption is
40%
with this non-optimised sample. It is possible that a sample that is optimised for the
133
8. Optical Response of Asymmetric Zigzag Bigratings
in-plane wavevector, k
x
(m
1
)
angular frequency, ω (rad s
1
)
0.2
0.4
0.6
0.8
1.0
0.2 0.4 0.6 0.8 1 1.2
× 10
7
2.5 3 3.5 4 4.5
× 10
15
+1
-1
0
+2
Figure 8.10: The dispersion of the SPP modes mapped as a function of unpolarised
light reflectivity on an asymmetric zigzag grating. The blue lines are the calculated
positions of the diffracted light lines. The colour scale indicates the absolute reflectivity.
coupling of light to the surface modes could provide 100% absorption of polarised light
and excitation of the SPP bands. This would be a highly efficient mechanism by which
to excite the same SPP mode along a surface, since on ordinary diffraction gratings,
50% of un-polarised incident light will not couple at
φ
= 0
. Plasmonic devices such
as SPP enhanced fibre Bragg gratings [
145
,
146
] could benefit from such an optimised
structure as conversion of TE incident light in to a SPP surface wave with TM character
could improve the coupling into TM fibre optic modes. SPP enhanced photo-detectors
[147] could also benefit from these relaxed polarisation constraints.
8.4 Band Structure of SPPs on an Asymmetric Zigzag
Grating
In chapter 7, we saw how a zigzag grating which possesses mirror symmetry has strict
constraints on the possible band structure of SPPs travelling along such a grating. It
was shown that the interaction of the
1
k
gx
and +1
k
gx
scattered SPPs is very weak
and, most strikingly, that the formation of a band-gap at the first BZ (the interaction
between the
1
k
gx
and +2
k
gx
scattered SPPs) was forbidden by the symmetry of the
charge distributions.
In the case of an asymmetric zigzag grating, both these conditions on the SPP band
structure are no longer true. Band-gaps are experimentally observed at both the high-
134
8. Optical Response of Asymmetric Zigzag Bigratings
(a) Symmetric (b) Asymmetric
Figure 8.11: The iso-frequency contours at
λ
0
= 550 nm measured on: (a) a symmetric
zigzag from chapter 7; (b) An asymmetric zigzag grating. Along
k
x
= 0 two SPP bands
cross in both cases, one scattered from
k
gx
, and one from +
k
gx
. The red square inset
in (b) shows the magnified region of the band gap, with the red arrows showing that
the direction of the group velocity contains no k
x
component.
symmetry points: at
k
x
= 0 and at the first BZ boundary,
k
x
=
π
gx
. The following
section outlines the experimental observations of these band-gaps on an asymmetric
zigzag grating, and offers a discussion on the origin of these SPP energy gaps. Further
to these results, evidence is found that the combination of highly anisotropic dispersion
of SPPs and the allowed formation of band-gaps may forbid the propagation of SPPs in
all directions for a range of frequencies, forming a full plasmonic band-gap.
8.4.1 Band Gaps at k
x
= 0
The interaction of the
±k
gx
SPP modes at
k
x
= 0 requires a sufficiently large Bragg
scattering amplitude of the SPPs so that they might interfere strongly and form a
standing wave. The two possible SPP standing waves, which lay along the zigzag
surface and not over the grooves, will possess energetically different arrangements of
charge density. These standing waves have different frequencies, and between these
two frequencies SPP propagation is forbidden; a plasmonic band gap exists. This
second point is also true of symmetric zigzags, covered in chapter 7, but SPPs on these
symmetric gratings are only weakly Bragg scattered, as the surface functions contains
no significant even components to couple the SPP modes together. In the asymmetric
zigzag case, however, the surface function of the asymmetric zigzag pattern introduces
higher harmonics to the grating surface profile, both odd and even. These harmonics
provide a direct coupling mechanism of 2
k
gx
for the
±k
gx
SPP modes meeting at
k
x
= 0,
135
8. Optical Response of Asymmetric Zigzag Bigratings
and consequently a plasmonic band gap may form.
This is observed for an asymmetric silver zigzag grating fabricated as outlined in
section 8.2, using imaging scatterometry to obtain the iso-frequency contours of the SPP
modes. For the case of an illuminating wavelength of
λ
0
= 550 nm, the obtained contour
is shown in figure 8.11(b). For comparison, the scattergram of the symmetric zigzag
grating from chapter 7 is also included in figure 8.11(a). Both experimental maps use the
polarisation which gives the greatest contrast of the contours over the entire momentum
range. Due to SPPs being excited with either polarisation in a asymmetric case, the
iso-frequency contour in figure 8.11(b) is recorded using unpolarised light, while the
symmetric zigzag is recorded using plane polarisers. These experimental results show
that at the intersection of the two scattered SPP contours at
k
x
= 0, the SPP contours
are split in to two distinct bands for the asymmetric case. The group velocity of the
SPP modes is
v
g
=
ω
(
k
), and so the local group velocity direction is determined on
these diagrams as orthogonal to the local contour. At
k
x
= 0, there is no component
of
v
g
in the
x
direction, since in this direction standing waves have formed. Compare
this to the result shown in figure 8.11(a), for the zigzag grating from chapter 7. These
SPP bands pass through each other unperturbed, having only weak multiple-scattering
processes available to them by which to couple.
The surface of our sample provides the strong scattering and so stronger interaction
of these scattered SPP modes by a combination of a larger zigzag amplitude and the
asymmetery introduced by offsetting an apex by δ = 150 nm.
8.4.2 Band Gaps at the 1
st
BZ Boundary
The lack of a band-gap at the first BZ for SPPs on symmetric zigzag gratings detailed
in chapter 7 is a result of the two possible standing wave solutions being degenerate in
energy. This is because the two possible arrangements for a standing wave of charge
density on a zigzag surface lie in identical electromagnetic environments. Since there is
no difference in energy between the two states, no band gap is observed.
In the experimentally mapped dispersion of the asymmetric zigzag grating, shown
in figure 8.10, a clear band-gap has now opened at the first BZ, where the
1
k
gx
and +2
k
gx
scattered SPPs cross. In this angular frequency range of 3
.
9
×
10
15
rad
s
1
to 4.2 × 10
15
rad s
1
, there is no coupling of light to either SPP, meaning no SPP
propagation occurs within this frequency gap for this orientation of the grating. The
gradient of the SPP contours falls to zero at the BZ, meaning that the SPPs on the
upper and lower frequency band edges have zero group velocity, and are standing waves.
This band gap is a consequence of the broken mirror symmetry of the diffraction
grating, and may be demonstrated using a FEM model. In figure 8.12 we show the
136
8. Optical Response of Asymmetric Zigzag Bigratings
x (nm)
y (nm)
0 300 600
0 75 150
in-plane wavevector, k
x
(m
1
)
angular frequency, ω (rad s
1
)
5.1 5.15 5.2 5.25 5.3 5.35
× 10
6
× 10
15
2.8 2.81 2.82 2.83 2.84
BZ
(a) Symmetric
x (nm)
y (nm)
0 300 600
0 75 150
in-plane wavevector, k
x
(m
1
)
angular frequency, ω (rad s
1
)
5.1 5.15 5.2 5.25 5.3 5.35
× 10
6
× 10
15
2.68 2.69 2.7 2.71
BZ
(b) Symmetric
x (nm)
y (nm)
0 300 600
0 75 150
0 150
in-plane wavevector, k
x
(m
1
)
angular frequency, ω (rad s
1
)
5.1 5.15 5.2 5.25 5.3 5.35
× 10
6
× 10
15
2.79 2.8 2.81 2.82 2.83 2.84
BZ
(c) Asymmetric
Figure 8.12: The calculated eigenmodes for the crossing of the
1
k
gx
and +2
k
gx
scattered SPPs at the first BZ for the cases of: (a) a symmetric zigzag; (b) a symmetric,
high-amplitude zigzag and; (c) an asymmetric zigzag. The unit cell for each is shown
above the modelled dispersion plots.
modelled dispersion for the
1
k
gx
and +2
k
gx
SPP crossing at the first BZ. The model
used is an eigenmode calculation using the FEM modelling techniques described earlier.
Three dispersions are shown, each for a different zigzag grating.
Figure 8.12(a) shows the dispersion at the BZ boundary for a typical symmetric
zigzag grating. As detailed in chapter 7, band-gaps are forbidden to form at this BZ
boundary by the symmetry considerations of the allowed charge distributions. As such,
the SPP dispersion is unperturbed, and the
1
k
gx
and +2
k
gx
scattered SPPs pass
through each other without interacting. This is also the case in figure 8.12(b), where
the symmetric zigzag amplitude has been increased by 150 nm. This unit cell maintains
the mirror symmetry of the zigzag grating and so, once again, formation of a band gap
is forbidden. However, by offsetting the zigzag grating, as modelled in figure 8.12(c),
the formation of a band gap is immediately achieved in the modelled SPP dispersion.
The group velocity (the gradient of the SPP dispersion curves) falls to zero at the BZ
boundary, and a region of forbidden SPP propagation opens up.
The formation of a band gap at the first BZ boundary is a consequence of the SPP
standing waves differing in energy. Figure 8.13 shows the modelled magnitude of electric
field plotted in the xz plane for the low and high energy standing wave solutions.
These two standing wave solutions have three ‘hot-spots’ of electromagnetic field
137
8. Optical Response of Asymmetric Zigzag Bigratings
x (nm)
z (nm)
0.01
0.02
0.03
0.04
0.05
0.06
0 300 600
0 100 200 300 400
(a) low energy
x (nm)
z (nm)
0.05
0.10
0.15
0.20
0.25
0 300 600
0 100 200 300 400
(b) high energy
Figure 8.13: Field plots of
|E|
(colourplot) a for the (a) low energy and (b) high energy
standing waves at the first BZ. The cross section is taken at y = 75 nm.
per unit cell, equating to a wave with a wavevector 3
k
gx
/
2, as you would expect for
a SPP standing wave at the first BZ formed between the
1
k
gx
and +2
k
gx
scattered
SPPs . The difference in energy between the two modes is easily identifiable by the
decay lengths of the electric fields. In the low energy case, the SPP dispersion is pushed
down in energy, away from the light line. Consequently, the electric field profile is more
‘plasmon-like’, confined closer to the surface than the high energy solution, which is
more ‘photon-like’, having been pushed up in frequency, towards the diffracted light
line.
The origin of this difference in energy between the two standing wave states is due
to the different arrangements of surface charge along the zigzag. This is demonstrated
in the modelled magnitude of electric field plotted in a plane just above the zigzag,
displayed in figure 8.14.
In this figure, the two standing wave solutions are shifted spatially by a quarter
phase lag along the grating. This positions the nodes and anti-nodes of the standing
wave as shown in the figure, with the low energy solution placing the field extrema
close to the apexes of the zigzag, and placing the high field points for the high energy
solution along the edges of the zigzag. Without any mirror symmetry of the zigzag
grating, these two arrangements for the standing wave surface charge are different, and
so an energy difference exists between them. As a consequence, an SPP band gap now
opens at the first BZ boundary.
8.4.2.1 Coupling of Light to the Band Edges
Figure 8.15 shows the experimentally mapped band gap at the first BZ boundary for
the three polarisation cases; TE, TM and average polarised light. It is seen that in
138
8. Optical Response of Asymmetric Zigzag Bigratings
x (nm)
y (nm)
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0 300 600
0 75 150
(a) low energy
x (nm)
y (nm)
0.1
0.2
0.3
0.4
0.5
0 300 600
0 75 150
(b) high energy
Figure 8.14: Field plots of
|E|
(colourplot) and
ˆ
E
(arrows) for: (a) the low energy and;
(b) high energy standing waves at the first BZ. The cross sectional height is 1 nm above
the grooves at 41 nm.
in-plane wavevector, k
x
(m
1
)
angular frequency, ω (rad s
1
)
0.2
0.4
0.6
0.8
1.0
0.3 0.4 0.5 0.6 0.7
× 10
7
3.6 3.8 4 4.2 4.4
× 10
15
(a) TE
in-plane wavevector, k
x
(m
1
)
angular frequency, ω (rad s
1
)
0.2
0.4
0.6
0.8
1.0
0.3 0.4 0.5 0.6 0.7
× 10
7
3.6 3.8 4 4.2 4.4
× 10
15
(b) TM
in-plane wavevector, k
x
(m
1
)
angular frequency, ω (rad s
1
)
0.2
0.4
0.6
0.8
1.0
0.3 0.4 0.5 0.6 0.7
× 10
7
3.6 3.8 4 4.2 4.4
× 10
15
(c) Average
Figure 8.15: Experimental reflectivity colour plots mapping the dispersion of the band
gap around the 1st BZ, for incident (a) TE polarised light, (b) TM polarised light and
(c) average polarisation. The red dotted line indicates the position of the BZ boundary.
all three cases, the lower energy band edge is coupled to by both TE and TM light
relatively strongly with a dark band of
20% reflectivity, while the higher-frequency
band edge is never coupled to as strongly, with a reflectivity of
90%, absorbing only
10% of the incident light.
To determine why light will preferentially couple to the lower frequency mode, we
may consider the overlap integral of the fields for both cases. The overlap integral
gives a quantitive measure of how much a driving electric field will match that of the
resonance, and hence allow us to determine relative coupling strength between the two
modes. The overlap integral is defined in a plane as [148],
ζ =
|
R
E
1
E
2
dA|
2
R
|E
1
|
2
dA
R
|E
2
|
2
dA
(8.8)
139
8. Optical Response of Asymmetric Zigzag Bigratings
E
1
and
E
2
are the two electric fields to consider, in our case the SPP eigenmode and
the incident plane wave. A plane of area
A
integrated over an area equal to the plane of
incidence, in this case (
φ
= 0), the
xz
plane. We shall use field profiles for the resonant
standing waves from the calculated FEM model and make several simplifications to the
overlap expression for
ζ
. Firstly, we note that the
z
dependence for the electric field
of a SPP has the same functional form in each case, an exponential in the
z
direction.
Immediately then, we can simplify the integral to a one-dimensional line integral in
x
,
as the normalized contributions from the
E
z
of the SPP and the light field overlap will
be equal in both cases. Secondly, we take the sum of the
E
x
field along
y
to account for
the whole unit cell. Since this is now a numerical calculation, the expression for the
simplified overlap, ζ
?
is,
ζ
?
=
|
P
λ
x=0
E
x
SP P
E
0
x
|
2
P
λ
x=0
|E
x
SP P
|
2
P
λ
x=0
|E
0
x
|
2
(8.9)
The data for
E
x
SP P
(
x
) are extracted from the numerical simulation shown in figure
8.14, for a region in the grooves at
z
= 20 nm. At the first BZ boundary, the incident
light,
E
0
x
(
x
) has an in-plane wavevector equal to
k
x
=
k
gx
/
2, and so the expression used
for E
0
x
is,
E
0
x
= sin
k
gx
x
2
+ ψ
(8.10)
with
ψ
equal to the phase of the wave at the surface. Since the band-gap under
consideration is not for normal incidence, the incident electric field propagates in the
x
direction, which we can represent as a change in
ψ
. At certain values of
ψ
, optimal
coupling will occur and at 90
in phase to this, no coupling will occur. When considering
the overlap integral for each standing wave case, we optimise the value of
ψ
to give the
largest value of
ζ
?
, corresponding to a phase at which the light is optimally matched to
the standing-wave fields. The functional forms for both
E
0
x
and
E
x
SP P
, for both band
edge solutions are shown in figure 8.16.
It is found that for the low energy solution,
ζ
?
= 0
.
0212(4
d.p.
), while for the high
energy solution
ζ
?
+
= 0
.
0000(4
d.p.
). The overlap of the impinging
E
x
is far greater for
the lower energy solution by comparison with the high energy solution, which to a fair
approximation is zero. Returning to figure 8.16, the physical reason for this preferred
coupling is now clear. In the low energy case,
E
x
is compressed (and so enhanced) in
region 2 (450 nm
< x <
600 nm). This region correspond to
E
x
originating from just
right of the zig-zag apex crossing the grooves to the lower apex edge at
x
= 600 nm,
whereas for
E
+
x
, the field is reasonably uniformly distributed across the zigzag. The
asymmetrically enhanced field in the
E
x
case adds a
k
gx
/
2 component to the SPP field,
140
8. Optical Response of Asymmetric Zigzag Bigratings
-1.0 -0.5 0.0 0.5 1.0
x (m)
E
x
(a.u.)
0 2 × 10
-7
4 × 10
-7
6 × 10
-7
8 × 10
-7
1 × 10
-6
1.2 × 10
-6
(a) low energy
-1.0 -0.5 0.0 0.5 1.0
x (m)
E
x
(a.u.)
0 2 × 10
-7
4 × 10
-7
6 × 10
-7
8 × 10
-7
1 × 10
-6
1.2 × 10
-6
(b) high energy
Figure 8.16: The (a) low energy and (b) high energy
x
-component of the SPP standing
wave electric field (black,
E
SP P
) and incident field (red,
E
x
) varying across two unit
cells of the asymmetric zigzag grating (light grey shapes indicate the zigzag in the
xy
plane).
allowing it to couple strongly to the light, which itself has a
k
gx
/
2 component at the BZ
boundary. Similar explanations exist for the coupling for band edges on surface relief
blazed gratings [85, 114] explained in chapter 4.
8.4.2.2 A Full Surface Plasmon Band Gap
SPP propagation on a zigzag grating is naturally anisotropic with respect to propagation
direction, due to the large band gaps which can form when the SPPs run over the
grooves, and the relativity small (or forbidden) band gaps that form when the SPPs
run along the zigzags. In chapter 7, it was shown that this anisotropy can be so large
that a zigzag grating may form the basis of a SPP collimating device, with the group
velocity of SPP modes solely in one direction. The asymmetric zigzag also shows this
increasing anisotropy with frequency, as shown in figure 8.17. As the frequency of the
light is increased, the SPP contours lie further from the zero-order circular SPP cones,
and become more ellipsoidal. These contours also split due to band-gaps along the
k
x
= 0 symmetry plane.
The anisotropic propagation combined with the ability for SPPs to form band gaps
at the 1
st
BZ boundary lead to a mechanism by which we may forbid the propagation
of SPPs in all directions along the grating surface, forming a full plasmonic band gap.
When the dispersion of the SPPs has become sufficiently anisotropic so that the
group velocity of the SPPs are in a single direction, the SPP contours intersect the
first BZ at
π
gx
with momentum only in the
x
direction. This was observed before
in the symmetric zigzag case in chapter 7: figure 7.23, where the SPP contours pass
141
8. Optical Response of Asymmetric Zigzag Bigratings
(a) 650 nm (b) 600 nm (c) 580 nm
(d) 550 nm (e) 500 nm (f) 450 nm
Figure 8.17: Iso-frequency contours of an asymmetric zigzag grating for a range of wavelengths, measured using the reflectivity of
unpolarised light in the imaging scatterometer. The blue circles indicate calculated diffraction edges.
142
8. Optical Response of Asymmetric Zigzag Bigratings
in-plane wavevector, k
x
(m
1
)
angular frequency, ω (rad s
1
)
0.2
0.4
0.6
0.8
1.0
0 0.2 0.4 0.6 0.8 1 1.2
× 10
7
2.5 3 3.5 4 4.5
× 10
15
+1
-1
0
+2
(a) (b)
Figure 8.18: (a) The dispersion of the SPP modes on an asymmetric zigzag grating for
φ
= 0
mapped using the reflectivity of unpolarised light. Included are the wavelength
slices corresponding to the iso-frequency scattergrams in figure 8.17, ranging from low
frequency (red, 650 nm) to high (blue, 450 nm ), (b) The scattergram for
λ
0
= 450 nm,
corresponding to the highest frequency slice in (a).
through the BZ boundary with the contour parallel to the boundary. In the asymmetric
zigzag grating case, the SPP iso-frequency contours still approach the BZ boundary
approximately parallel, but now form a band gap at the boundary. This means that the
SPP band gap at this boundary forms at all values of momentum simultaneously, and
so a frequency band over which all SPP propagation is forbidden is opened. This is a
full plasmonic band gap. Figure 8.18 shows an iso-frequency contour for
λ
0
= 450 nm
close to the forbidden frequency region, and shows minimal coupling for light to SPPs
across the entire light cone. This iso-frequency contour is not at the exact frequency of
the observed band gap, shown in the corresponding dispersion plot in figure 8.18, due
to experimental limitations. The poorly coupled SPP contours in the scattergram also
show stronger (yet still weak) coupling towards the edges of momentum-space, and it is
clear that the contours are, in fact, slightly curved. This shows that the dispersion is
not as anisotropic as would be necessary for the contour to intersect the BZ boundary
simultaneously. However, these results demonstrate in principle the engineering of a full
plasmonic band gap by using a sub-wavelength periodicity to introduce large anisotropy
in SPP propagation and then the ability for such gratings to form band gaps at the BZ
in all directions simultaneously. Optimising the sample and experimentally measuring
the forbidden frequency band is left as future work.
143
8. Optical Response of Asymmetric Zigzag Bigratings
8.5 Conclusions
We have demonstrated in this chapter that manipulation of the surface symmetry on a
zigzag grating can lead to some novel optical effects.
The coupling of any polarisation to SPP modes on such gratings was explained and
experimentally verified in section 8.3, with both TE and TM polarised light resonantly
driving the same SPP modes. The coupling strength was still preferentially to TE or
TM polarised light depending on the order of diffraction involved, but the optimisation
of this coupling might be possible using a different broken symmetry geometry.
The formation of band gaps on these gratings has also been measured, for both
planes of symmetry accessible to experiment for this sample. The energy difference
between these standing waves arises from the energetically dissimilar arrangements of
charge along the asymmetric zigzag. The coupling of light to these standing wave states
is found to depend on the symmetry of the surface, analogous to how the phase of
Fourier components determines the light coupling on surface-relief gratings. This is
explained in depth in section 8.4.
Finally, the potential for this structure to provide a full surface plasmon band gap
is explored, with the combination of the highly anisotropic SPP dispersion and the
formation of band-gaps at the BZ boundary showing promise for the engineering of a
full surface plasmon band-gap device.
144
Chapter 9
Conclusions
9.1 Summary of Thesis
This thesis details original experimental investigations on the excitation of SPPs along
metallic diffraction grating surfaces. Broadly, these investigations can be divided in to
two main categories: the investigation of SPPs on traditional ‘crossed’ bigratings with
novel symmetries, and the introduction of a new type of diffraction grating supporting
SPPs, the ‘zigzag’ grating.
The work on the traditional ‘crossed gratings’ is presented in chapters 5 & 6, exploring
gratings with symmetries that have previously received little attention in the literature.
Chapter 5 experimentally explores the dispersion and coupling of SPPs on a rectangular
bigrating. The groove profiles of such bigratings are described as a Fourier expansion
of the constituent gratings, and this is used to explain the experimentally observed
scattering and interaction of the supported SPP modes. Surface plasmon band-gaps
are seen to occur at the boundaries of the rectangular BZ and are recorded using the
novel technique of imaging scatterometry. This measurement technique for acquiring
the iso-frequency contours of SPPs in
k
-space is original to this thesis, and detailed in
chapter 4. The measured SPP contours at the BZ boundary are reproduced using FEM
modelling, showing good agreement with experiment.
Chapter 5 also shows that by increasing the depth of a constituent grating, the SPP
travelling in the orthogonal direction can be slowed in group velocity. Experimental
results also show that the SPP iso-frequency contour can be shaped by changing the
available grating harmonics and hence the strength of scattering amplitudes. This
is demonstrated experimentally as a mechanism by which to design anisotropic SPP
propagation, which could be useful for the development of SPP surface optics.
Chapter 6 investigates SPPs on gratings with the lowest symmetry of all the 2D
Bravais lattices, oblique bigratings. SPP mediated polarisation conversion is observed
145
9. Conclusions
on these gratings due to the broken mirror symmetry of the surface. Measurements of
the dispersion also show evidence that SPPs may undergo self-coupling. This is the
mechanism of a propagating SPP mode being able to resonantly drive another, different,
SPP mode. This is a possibility providing the two modes do not travel orthogonally to
one another.
The iso-frequency contours of these SPPs are mapped experimentally using imaging
scatterometry, and it is found that band-gaps do not necessarily occur at the conventional
definition of the BZ boundary. A discussion of the symmetry considerations on such a
grating is used to explain why this is the case, concluding that since the BZ boundary
is itself not a contour of high-symmetry for an oblique lattice, there is no condition
for the band-gaps to form at this arbitrary boundary. There are, however, isolated
high-symmetry points along the BZ boundary, and when the plane of incidence intersects
these unique points, it is experimentally observed that SPP band-gaps do still occur.
The final two experimental chapters introduce a new type of diffraction grating: the
zigzag grating. These are gratings where a diffractive periodicity has been introduced
by ‘zigzagging’ a set of sub-wavelength (non-diffracting) grooves in a metal surface. The
electric field of plane polarised light incident on such a zigzag grating will intersect the
surface regardless of the polarisation angle, inducing surface charge. This results in
SPP excitation on these gratings with either TE or TM polarised light.
Chapter 7 details such a zigzag grating that possesses a single mirror plane. The
symmetry of this zigzag structure leads to the observation that TE polarised light
excites SPPs scattered by odd-order grating vectors, while TM polarised light excites
SPPs scattered by even-orders. This separation of SPP diffracted orders by polari-
sation selectivity is explained using a simple theoretical treatment, and is observed
experimentally.
A second consequence of this symmetric zigzag grating is that the standing surface
wave states at the first BZ boundary are found to be degenerate in energy. By modelling
the system using the FEM, it is found that the two possible SPP standing waves inhabit
identical electromagnetic environments, and so no energy difference exists between them
and no bad-gap may form. Experimental observations confirm this lack of SPP band-gap
at the first BZ boundary.
The final results of chapter 7 show that SPP band-gaps associated with the short
sub-wavelength pitch cause the observed SPP iso-frequency contours to deform. So
large is this perturbation that at high frequencies the SPP contours are flat, and the
SPP’s group velocity is constrained to be parallel to the long-pitch direction, irrespective
of incident angle of the coupling light. This, combined with the forbidden band-gaps,
makes these zigzag gratings excellent candidates for surface wave collimation devices.
The final experimental results of this thesis are shown chapter 8 and pertain to the
146
9. Conclusions
SPPs excited along a zigzag grating which possesses no mirror symmetry. This leads
to the experimental observation that either TE or TM polarised light may excite the
same SPP modes. Band-gaps may now form, as the asymmetric zigzag surface leads
to different energetic arrangements of surface charge for the different standing wave
states. The anisotropic propagation of SPPs on such a surface observed in chapter 7
is also observed for these asymmetric zigzag gratings, which, when combined with the
large SPP band-gaps, leads to the formation of a full SPP band-gap, for which SPP
propagation is forbidden in all directions.
9.2 Future Work
Throughout this thesis we have demonstrated some novel optical effects related to SPPs
on metal gratings, and this provides a wealth of possible future work.
The manipulation of SPP band structures presented using rectangular bigratings
in chapters 5, 7 and 8 may provide useful tools in the design of SPP surface optics,
including lenses, collimators and perhaps even negative index devices. Smoothly varying
the depths along such gratings would change the effective mode index along the surface,
and be used to guide or direct SPPs.
The use of sub-wavelength grooves shaped along their length to provide diffractive
coupling to SPPs has a large potential for many interesting future studies. The use of
the grating in chapter 7 as a collimating device for surface waves would be an excellent
avenue for investigation. A schematic of a possible experimental arrangement is shown
in figure 9.1. In this figure, a SPP point source (red dot) excites and couples to SPPs
on a planar metal surface. The use of a near-field optical microscopy (SNOM) tip or
a small point scatterer illuminated with a laser could provide near-field coupling into
a SPP wave which, in the absence of the grating, will propagate outwards radially.
With the addition of a grating similar to the one presented in chapter 7, the allowed
momentum states of the SPP are constrained to propagate in a single direction only,
causing collimation of the surface wave. The distance between the grating and the
point source would control the final (presumably Gaussian) SPP beam width. This
collimation could then be observed by measuring the electric field across the surface
using SNOM, or by using leaky radiation microscopy [61] to image the beam.
The potential exists, by altering the zigzag period, the sub-wavelength period or the
depth of the sub-wavelength grooves, to allow for the onset of flat SPP contours at a
lower frequency than that found in chapter 7, possibly providing a device which can
collimate SPP waves over a broad range of frequencies.
Another potential device that could incorporate a zigzag diffraction grating is an
SPP mediated light source. Metal-Insulator-Metal (MIM) devices with the top metal
147
9. Conclusions
SPP Source
k
SP P
k
SP P
Zigzag grating
Figure 9.1: A possible experimental arrangement for the observation of collimated
surface waves using zigzag gratings.
surface corrugated to form a diffraction grating have been shown to emit light when
driven with a suitable voltage [
149
]. This process is mediated by the excitation of SPPs
by electrons followed by the out-coupling of these SPPs to light by the diffraction grating.
Replacing the top surface diffraction grating with the symmetric zigzag grating described
in chapter 7 would result in the emission of TE polarised light into the
±
1
, ±
3
, ±
5
, ...
diffracted orders and also the emission of TM polarised light into the
±
2
, ±
4
, ...
orders.
The device would be a polarisation directing/separating light source. The use of an
asymmetric zigzag as detailed in chapter 8 would result in the emission of unpolarised
light into the diffracted orders.
The asymmetric zigzag in chapter 8 also provides some exciting possibilities for other
future work. Investigation into the effect of different asymmetric patterns would most
likely lead to effects analogous to changing the phase of a 2
k
g
component on a traditional
surface-relief grating. If a device could be constructed where the unpolarised light could
couple to the SPP standing wave states (and so a region of high density of SPP states)
in the upper band edge (as opposed to the preferential coupling to the lower band
edge observed in chapter 8), one could envisage a device with great potential in SPP
enhanced lasers or SPP enhanced solar cells. In such devices, it is desirable to couple
the SPP into a gain medium coating the surface, which favours the upper-band edge
of SPP band-gaps since these fields are more ‘photon-like’ extending further into the
surround gain medium that the ‘plasmon-like’ lower-band edge. The advantage of using
an optimised asymmetric zigzag is that the same SPP modes can be coupled to with
any incident polarisation, possibly improving efficiency. The polarisation insensitivity
148
9. Conclusions
to the generation of the locally TM polarised SPP waves could also prove useful for the
efficient coupling of light into optical fibres.
149
Publications
Journal Articles in Print
Surface Plasmons on Zig-Zag Gratings, T. J. Constant, T. Taphouse, H. J. Rance,
S. C. Kitson, A. P. Hibbins, and J. R. Sambles. Opt. Express
20
, 23921–23926
(2012).
Surface Waves at Microwave Frequencies Excited on a Zigzag Metasurface, H.
J. Rance, T. J. Constant, A. P. Hibbins, and J. R. Sambles. Phys. Rev. B
86
,
125144 (2012).
Journal Articles in Preparation
Mapping Surface Plasmon Iso-Frequency Contours Using Imaging Scatterometry,
T. J. Constant, A. P. Hibbins, A. P. Lethbridge, E. K. Stone, J. R. Sambles and
P. Vukusic. (2013).
Poster Presentations
Direct Imaging of Surface Plasmon Band-gaps on zig-zag gratings, T. J. Con-
stant, A. P. Hibbins and J. R. Sambles. 4
th
International Topical Meeting on
Nanophotonics and Metamaterials, Seefield, Tirol, Austria. (January 2013)
Zig-Zag Gratings, T. J. Constant, A. P. Hibbins and J. R. Sambles. 3
rd
Inter-
national Topical Meeting on Nanophotonics and Metamaterials, Seefeld, Tirol,
Austria. (January 2011)
Surface Plasmons on Rectangular Bi-Gratings, T. J. Constant, A. P. Hibbins and
J. R. Sambles. The Institute of Physics: Plasmonics UK Meeting, Institute of
Physics, London, United Kingdom. (May 2010)
150
9. Conclusions
Surface Plasmons on Rectangular Bi-Gratings, T. J. Constant, A. P. Hibbins and J.
R. Sambles. The Royal Society, Theo Murphy International Scientific Meeting on
Metallic Metamaterials and Plasmonics, Chicheley Hall, United Kingdom. (June
2010)
Future Publications
Other publications arising from this thesis and intended for submission in the near future
include: a paper on the self-collimation characteristics of zigzag gratings presented in
chapter 7, and a paper on the excitation of SPPs on the asymmetric zigzag grating
geometry in chapter 8.
151
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